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HIGH SCHOOL DEPARTMENT
Quarter 1 1st Se Semest mest er
Prepared by: Senior High School Teacher
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
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HIGH SCHOOL DEPARTMENT Table of Contents
Lesson 1 Functions ……………………………………………………………………………………………..3
Lesson 2 Rational Functions …………………………………………………………………………………..16
Lesson 3 Intercepts, Zeroes, and Asymptotes of Rational Functions……………...…………………………………………………………….……………….. 28
Lesson 4 Inverse Functions ……………………………………………………………………..……………..38
Lesson 5 Exponential Functions ………………………………………………………………..……………..43
Lesson 6 Logarithmic Functions ……………………………………………………………….……………..50
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HIGH SCHOOL DEPARTMENT
Learning Objectives: a. recall the concepts of relations and functions; b. define and explain functional relationship as a mathematical model of situation; and c. represent real-life situations using functions, including piece-wise function.
Content Standard:
The learners demonstrate understanding of key concepts of ffunctions. unctions.
Performance Standard:
The learners are able to accurately construct mathematical models to represent real-life situations using functions.
Overview Function is defined as “a relation in which each element of the domain corresponds to exactly one element of the range.” In this module, we will represent
real-life situations using functions, evaluate functions, and perform operations on functions.
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HIGH SCHOOL DEPARTMENT Lesson Discussion: Function and Relation
A relation is any set of ordered pairs. The set of all first elements of the ordered pairs is called the domain of the relation, and the set of all second elements is called the range. A function is a relation or rule of correspondence between two elements (domain and range) such that each element in the domain corresponds to exactly one element in the range. Example Given the following ordered pairs, which relations are functions? A = {(1,2), (2,3), (3,4), (4,5)} B = {(3,3), (4,4), (5,5), (6,6)} C = {(1,0), (0, 1), (-1,0), ( -1,0), (0,-1)} D = {(a,b), (b, c), (c,d), (a,d)} Solution The relations A and B are functions because each element in the domain corresponds to a unique element in the range. A = {(1,2), (2,3), (3,4), (4,5)} B = {(3,3), (4,4), (5,5), (6,6)}
Meanwhile, relations C and D are not functions because they contain ordered pairs with the same domain [C = (0,1) and (0,-1), D = (a,b) and (a,d)]. C = {(1,0), (0, 1), (-1,0), ( -1,0), (0,-1)} D = {(a,b), (b, c), (c,d), (a,d)} Example How about from the given table of values, which relation shows a function? A. X Y
1 2
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
2 4
3 6
4 8
5 10
6 12
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HIGH SCHOOL DEPARTMENT B. X
4
-3
1
2
5
Y
-5
-2
-2 -2
-2
0
X
0
-1
4
2
-1
Y
3
4
0
-1
1
C.
Solution: A and B are functions since all the values of x corresponds to exactly one value of y. Unlike table C, where -1 corresponds to two values, 4 and 1. X
0
-1
4
2
-1
Y
3
4
0
-1
1
Example We can also identify a function given a diagram. On the following mapping diagrams, which do you think represent functions? A.
C.
B.
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HIGH SCHOOL DEPARTMENT Solution The relations A and C are ar e functions because each element in the domain corresponds to a unique element in the range.
However, B is a mere relation and not function because there is a domain which corresponds to more than one range.
A relation between two sets of numbers can be illustrate by graph in the Cartesian plane, and that a function passes the vertical line test. A graph of a relation is a function if any vertical line drawn passing through the graph intersects it at exactly one point. Example Using the vertical line test, can you identify the graph/s of function?
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HIGH SCHOOL DEPARTMENT A and C are graphs of functions while B and D are not because they do not pass the vertical line test.
Note: we can represent represent functions iinn different ways. It can be represented through words, tables, mappin mappings, gs, equations and graphs.
Piecewise Functions
A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. formula 1 if x is in domain 1
f(x) =
formula 2 if x is in domain 2 formula 3 if x is in domain 3
Example 1. A user is charged ₱250.00 monthly for a particular mobile plan, which includes 200 free text monthly cost for text messages. Messages in excess of 200 are charged ₱1.00 each. Re present the monthly messaging using the function t(m), where m is the number of messages sent in a month.
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HIGH SCHOOL DEPARTMENT Solution:
≤
t (m) = 250 if 0 0 < m 200
Fo Forr ssen endi dinn mess messaa es of no nott exc excee eedi dinn 20 200 0
(250 + m) if m > 200
In case the the messa messa es sent were were more more than 200 200
2. The cost of hiring a catering service to serve food for a party is ₱250.00 per head for 50 persons or less, ₱200.00 per head for 51 to 100 persons, and ₱150.00 per head for more than 100. Represent
the total cost as a piecewise function of the number of attendees to the party. Solution:
≤ ≤≤ >
c(h) = 250 if n 50
Cost for a service to at least 50 ersons
200 if 51 n 100
Cost for for a service to 51 to 1100 00 ersons
150 if n 100
Cost for for a service to m more ore than 100 ersons
Evaluation Function Evaluating function is the process of determining the value of the function at the number assigned to a
given variable. Just like in evaluating algebraic expressions, to evaluate function you just need to: a) replace each letter in the expression with the assigned value and operations in th thee expression us using ing the correct order of operations operations.. b) perform the operations Example : Given
= 22 4 4
, find the value of the function if
=3
.
Solution:
3 = 23 4
Substitute 3 for x in the function
Simplify expression right sidethe of the equationon . the
33 = =6 4 2
Answer: Given
= 22 4 4 3 = 2 = 3 7 ,
Example : Given
, find g(-3).
Solution:
3 3 = 33 7 3 3 = 339 7 3 3 = 27 7
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
Substitute -3 for x in the function Simplify the expression on the right side of the equation .
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3 3 = 34
Answer: Given
= 3 7
Example : Given
, g(-3) = 34
= 55 11 ℎ 1 , find
.
Solution:
ℎ 1 1 = 5ℎ 1 1 ℎ 1 1 = 5ℎ 5 1 ℎ 1 1 = 5ℎ 6 = 55 1 1 ℎ 1 1 = 5ℎ5ℎ 6
This time, you substitute (h+1) into the equation for x.
Answer: Given
,
Example 9: Given Solution:
Use the distributive property on the right side, and then combine like terms to simplify.
= √ 3 22
, find g(9).
9 = 392 3 92 9 = √ 272 2 72 9 = √25 9 = 5
Substitute 9 for x in the function.
Simplify the expression on the right side of the equation.
Answer: Given
= √ 3 2
Example : Given
, find g(9) = 5.
ℎ = + −
, find the value of the function if x = -5
Solution:
5 8 ℎ5 5 = 4254 20 8 ℎ5 5 = 104 ℎ5 5 = 1412
Substitute -5 for x in the function. Simplify the expression on the right side of the equation. (recall the concepts of the integers and simplifying fractions)
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ℎ5 5 = 67
Answer: Given
ℎ = + 5 = − ℎ5 = 2 = ,
Example: Evaluate Solution:
if
.
(32) = 2 (32) = 2 (32) = √8 (32) = √4 ∙ 2 Answer: Given
Simplify the expression on the right side of the equation. (get the cubed of 2 which is 8, then simplify)
(32) = 2√2
= 2 = 2√ 2 ℎ = ‖ ‖ 2 ,
Substitute for x in the function.
Example: Evaluation the function
where
⌊⌋
is the greatest integer function given
Solution:
ℎ2.4 = ‖2.4‖ 2 ℎ2.4 = 2 2 ℎ2.4 = 4
Answer: Given
ℎ = ‖‖ 2 ℎ2.4 = 4 = | 8| ,
Substitute 2.4 for x in the function. Simplify the expression on the right side of the equation. (remember that in the greatest integer function, value was
where
| 8|
means the absolute value of
8 =3 if
Solution:
3 = |3 8| 3 = |5 5|| 3 = 5
Answer: Given
= | 8| 3 = 5
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
,
.
rounded-off to the to the integer less thanreal thenumber number)
Example : Evaluate the function
= 2.4
Substitute 3 for x in the function. Simplify the expression on the right side of the equation. (remember that any number in the absolute value sign is always positive)
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HIGH SCHOOL DEPARTMENT Example : Evaluate the function
= 2 2 2 3 at
.
Solution:
2 33 = 22 3 3 22 3 2 2 3 3 = 4 12 9 4 6 2 2 3 = 4 12 9 4 6 2 2 3 = 4 12 4 9 6 2 2 3 3 = 4 16 16 117
Substitute 2x-3 for x in the function. Simplify the expression on the right sidee of the e uation sid uation..
Operation on Functions Definition. Let f and g be functions.
= = ▪ ▪ ▪ = ▪ ⁄ / / = / = 0 ° ° =
1. Their sum, denoted by , is the function denoted by 2. Their difference, denoted by , is the function denoted by 3. Their product, denoted by , is the function denoted by
.
.
.
, is the function denoted by , excluding the 4. Their quotient, denoted by values of x where . 5. The composite function denoted by . The process of obtaining a composite function is called function composition. Example : Given the functions:
= 5
= 22 11
ℎ = 2 9 5
Determine the following functions: a. ( f f + g) ( xx) b. ( f f - g) ( xx) c. ( f f ▪ g ) ( xx)
e. ( f f + g) (3) f. ( f f - g) (3) g. ( f f ▪ g ) (3)
d.
h.
3
Solution:
. ൫ ൯ = . ൫ ൯= = 52 1 == 3 54 2 1 = 6
Definition of addition of functions replace f(x) and g(x) by the given values combine like Definition of subtraction of functions replace terms f(x) and g(x) by the given values distribute the negative sign combine like terms
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. ൫ ▪൯ = ▪
Definition of multiplication of functions replace f(x) and g(x) by the given values
multiply the binomials
= 59▪ 25 1
. ቆℎቇ = ℎ
Definition of division of functions replace h(x) and g(x) by the given values factor the numerator
= +−−
= 25 21 1 = 25 21 1
cancel out the common factors
= 5 . ൫ ൯3 = 3 3 = 3 5 23 1 = 8 5 = 13
. ൫ ൯3 = 3 3 = 3 5 23 1 = 6
. ൫ ▪൯3 = 3▪3 = 3 5▪23 1 = 8▪5 = 40
. ቆℎቇ = ℎ33
= +−−
= 18 6 271 5 = 450
= 8
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HIGH SCHOOL DEPARTMENT Composition of functions:
In composition of functions, we will have a lot of substitutions. You learned in previous lesson that to evaluate a function, you will just substitute a certain number in all of the variables in the given function. Similarly, if a function is substituted to all variables in another function, f unction, you are performing a composition of functions to create another function. Some authors call this operation as “ function of functions”.
Example 16: Given Find
◦ ℎ ℎ
= 5 6 ℎ = 2 , and
Solution:
◦ℎ = ൫ℎ൯ ◦ ℎ = 2 = 5 6 22 = 2 2 5 2 6 = 4 4 5 10 6 = 9 20
Since
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
Definition of function composition Replace h(x)by x+2 Given Replace x by x+2 Perform the operations Combine similar terms
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HIGH SCHOOL DEPARTMENT Note: Note:
After you read and analyze Lesson 1, you may now proceed to Activity number 1. Activity 1 in General Mathematics Mathematics Lesson 1: Functions
Name: ______________________ __________________________________ ____________
Score: _______ ____________ _____
Grade & Section: _____________ _________________________ ____________
Date: _____________ _____________
I. Directions: Choose the letter of the best answer. Write your answer on the space provided before the number. ________1. Which Which of the fo following llowing is not true about function? a. Functions is composed of two quantities where one depends on the other. b. One-to-one function correspondence is a function. c. One-to-many Many-to-one correspondence d. correspondence is is aa function. function. ________2. In In a relation, w what hat do you cal calll the y valu values es or the outpu output? t? a. b. c. d.
Piecewise Range Domain Independent
________3. In In this table, what is the ddomain omain of th thee function? X Y
1 a
2 b
3 c
4 d
5 E
a. D:{2, 4, 6, 8, 10} b. D:{a, b, c, d, e} c. D:{1, 2, 3, 4, 5} d. y ={1, 2, 3, 4, 5, a, b, c, d} ________4. A perso personn can encode 10 1000 00 w words ords in every hour of typing job. Wh Which ich ooff the following expresses the total words W as as a function of the number n of hours that the person can encode? a. b.
= 101000 =
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c. d.
= 1000 = 101000
HIGH SCHOOL DEPARTMENT
________5. Eighty meters meters of fencing is available to enclose the rectangular garden of Mang Gustin. Give a function A that can represent the area that can be enclosed in terms of x. a. b. c. d.
= 40 40 = 80 80 = 40 = 80
II. Directions: Evaluate the following functions. 1.
= 33 5 5 2 , find
Solution:
2.
= 3|22|| 6 , find
Solution:
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HIGH SCHOOL DEPARTMENT III. Directions: Match column A with column B by writing the letter of the correct answer on the blank before each number. Column A Given:
= 2 = 55 3 3 = −+ = √ 5 5 = −
______1. ______2. ______3. ______4. ______5.
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
Column B
▪ ◦
a.
b.
7 √ 7 5 7 6 6 11
c. d.
+ − +
e.
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HIGH SCHOOL DEPARTMENT
Learning Objectives: a. represent real-life situations using rational function; b. distinguish rational function, rational equation, and rational inequality; and c. solve rational equations and inequalities. Content Standard:
The learners demonstrate understanding of key concepts of rational functions.
Performance Standard:
The learners are able to accurately formulate and solve real-life problems involving rational functions.
Overview A rational function is defined as “a function that is the ratio of two polynomials.” polynomia ls.” In this module we will solve rational equations, inequalities, inequalities, and
functions; represent rational function, determine the domain and range of a rational function; and graph rational function.
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HIGH SCHOOL DEPARTMENT Lesson Discussion: Rational Function
where and are polynomial functions and that such ≠ 0. The domain of is a set ofreal isnot zero. =numbers
A rational function,
is a function of the form
Example:
= +− , , ≠ 1 = , ≠ 0
1.
- Both numerator and denominator are polynom polynomial ial functions, denominator has restriction because it should not be equal to zero
2.
- The numerator 1 is a polynom polynomial ial function with a degree 0, the denominator is a polynomial function and it must be not equal to 0
Real-world relationship that can be modeled by rational functions: Example A car is to travel a distance of 70 kilometers. Express the velocity (v) as a function of travel of time (t ) in hours. Solution: Let us first show the relationship using a table. Remember that as time increases in travelling the velocity or the speed of a car will decrease Time (hours) Velocity (k/hr) Thus, the function
1 70
=
2 35
3 23.33
5 14
10 7
can represent v as a function of t
Rational Functions, Equations and Inequalit Inequalities ies
A rational expression can be described as a ratio or quotient of two polynomials. Consider the following algebraic expressions, determine whether they are rational or not and state the reason. 1. 2. 3.
−+; Rational expression because it is a ratio of two polynomials + ; Rational expression bbecause ecause 1 and x – 5 5 are polynomials −− √ −; Not a rational expression since the numerator is not polynomial +
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4.
HIGH SCHOOL DEPARTMENT ; Rational expression because the numerator x+5 and denominator 1 are polynomials
5 5
To determine the difference among rational function, rational equation and rational inequality study the table below:
Rational Equation
Definition
Rational Inequality
Rational Function
An equation involving An inequality involving A function of the form rational expression rational expressions where
4 = 1 1 5
Example
2 2 > 3 5
= 6 8 = 4
and are polynom polynomial ial functions and is not the zero function
Solving Rational Equations and Inequalities
1. Simplifying rational expression using the following steps.
2 24 2 22 2 2 21 2 22
Steps in simplifying rational expression
1. Factor the denominator of the rational expression
2. Cancel the common factor
3. Write the simplified rational expression
2. To multiply rational expressions you can do the following steps. Steps in multiplying rational expression 1. Factor out all possible common factors. 2. Multiply the numerators and denominators. 3. Cancel out all common factors. 4. Write the simplified rational expression
331 111 ∙ 3 1 1 ∙ 3 1 13 1 3 1 1 3 11 1 1 31 13 1 3 1 1 11 1 1 1 1 113 13 11 11
3. To add and subtract rational expressions with like denominators you can do the following steps.
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HIGH SCHOOL DEPARTMENT Steps in addition or subtraction of rational expressions with like denominators 1. the numerators of both expressions and keeping the common denominator 2. Combine like terms in the numerator. 3. Write the simplified rational expression.
5 11 3 4 81 38 84 58 8 8 5 838 3834 1 8
4. To add and subtract rational expressions with unlike denominators you can do the following steps. Steps in adding or subtracting rational expressions with unlike denominators 1. Factor the denominator of each fraction to help find the LCD. 2. Find the least common denominator (LCD) 3. Multiply each expression by its LCD 4. Write the simplified expression 5. Let the simplified expression as the numerator and the LCD as the denominator of the new fraction 6. Combine like terms and reduce the rational expression if you can. In this case, the rational expression cannot be simplified.
6 2 6 4 5 26 2 : : 2 222 2 3 22 3 6 2 2 2 266 2 23182222 242 3 2 24 2 222 33 2 2 2 2 33
Rational equation is an equation containing at least one rational expression with a polynomial in the
numerator and denominator. It can be used to solve a variety of problems that involve rates, times and work. Example: Rational Equation
1 You need to find the Least Common Denominator (LCD) 2 You need to multiply LCD to both sides of the equation to eliminate the fractions. 3 You simplify the resulting equation using the distributive property and then combine all like terms.
:2 2221 1 1 1 =2 1 122 : 1 = 1 ] 2 2 11 22[[ 2 2 1 1 2 1 1 22 4221 1 2== 1 1 2 3 = 0
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HIGH SCHOOL DEPARTMENT 4 You need to solve the simplified equation to find the value/s of x. In this case, we need to get the equation equal to zero and solve by factoring.
3 23 31==0 0 3==30 =11= 0 1 1 = 12 2 2 1 = 3 2 1 = 1 3 2 31 2 2 1 = 1 8 4 2 1 = 1 2 2 =1 12 2 11 11 1 = 12 2 1 = 1 0 0 2 0 = 12
5 Finally, you can now check each solution by Possible solutions are -3 and 1 substituting in the original equation and reject any extraneous root/s (which do not satisfy the equation). When
Solution
When
Not Solution
Rational inequality is an inequality which contains one or more rational expressions. It can be used in
engineering and production quality assurance as well as in businesses to control inventory, plan production lines, produce pricing models, and for shipping/w shipping/warehousing arehousing goods and materials. Example Rational Inequality
1 . Put the rational inequality in the general form where can be replaced by , and .
0 3 11 2 ≥ 0 3 112 1 ≥ 0 1 1 3 3 ≥ 0 1 1
>
2 Write the inequality into a single rational expression on the left-hand side
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HIGH SCHOOL DEPARTMENT 3 .Set the numerator and denominator Numerator : equal to zero and solve. The values you get are called critical values. Denominator:
3 3 == 30 1 1 == 10
4 Plotbreaking the critical on aline number line, thevalues number into intervals 5 Substitute critical values to the inequality to determine if the endpoints of the intervals in the solution should be When included or not.
= 3 −+ ≥ 2 −− − ≥ 2 − 2≥ 2 =1 + ≥ 2 − ≥ 2
3 11 ≥ 2 1
✓ (
When
= −3 is included in tthe he solution)
≥ 2 ☓ = 1 is not included in the solution
u
Illustration
3 11 ≥ 2 1
6 Select test values in each interval and substitute those values into the inequality
When
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
= 5
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HIGH SCHOOL DEPARTMENT
351 ≥ 2 51 14 ≥ 2 73 2.33 ≥62 = 5 = 1 311 ≥ 2 11 2 ≥ 2 2 1 ≥ 2 = 1
When
When
=3
331 31 ≥ 2 10 ≥ 2 2 5 ≥ 2 = 3
Note: a. If the test value makes the inequality TRUE, then the entire interval is a solution to the inequality. b. If the test value makes the inequality FALSE, then the entire interval is not a solution to the inequality. 7 Use interval notation to write the final answer.
∞,3 ∪ 1,∞ 1, ∞
Representation of Rational Functions F unctions
Rational function is written in the form of
=
. It should follow the following conditions; namely:
1. Both p(x) and q(x) are polynomial functions wherein it has no negative and fractional exponents. 2. The denominator or q(x) should not be equal to 0. 3. The domain of all values of x where q(x) ≠ 0.
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HIGH SCHOOL DEPARTMENT Example a. Define a formula for the average cost for every 5 days to stay in the resort f(x). To define the formula, use the formula in getting the average cost. Let the function be f(x). We can use the formula of getting an average. Average problems use the formula = / , where A= Average, X= cost, and s= number of days
Let f(x) represents the average cost per day and x represent the number in days. Note that ₱300,000.00 is a fixed price you need to pay plus the ₱700.00 per day divided by the number of days (x). We will have,
= 300000700
Observe that it is similar to the structure of our original formula. Note that you will be using a formula depending on the classification of problems given to you. b. For every 5 day stay in the resort, create a table of valu values es showing the average cost. Solution: Make a table of values with x-values at 0, 5, 10, 15, 20, 25, 30. x 0 5 10 15 20 25 30 y 0 20,965 41,930 62,895 83,860 104,825 125,790 From the table, we can observe that the average cost of stay decreases as the time increases. We can use a graph to determine if the points of this function follow a curve or a line. c. Graph the following points in the Cartesian plane.
By connecting the lines, we can clearly see that it follows a curve, thus a Rational Function. Domain and Range of a Rational Functions
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HIGH SCHOOL DEPARTMENT The domain of a function is the set of all values that the variable x can take while the range of a function is the set of all values that y or f(x) can take. If you can recall, r ecall, we can write the domain and range using different forms: 1. by roster format - this method enumerates the lists of all values in the set. Ex. The domain of r(t) are (1, 1.25, 1.5, 1.75, 2). 2. by set-builder form form or notation - for example, in numbers 10 to 20. you can say {x | x are even numbers from 10 to 20). The | is read as “such that.” Assuming that you also include odd numbers in the domain from 10 to 20, then, you can write the domain of the function D(x) as {x | x ϵ R, 10≤x≤20}, read as “x such that x is an element of a real number wherein x is greater than or equal to 10 but less than or eq ual ual to 20.”
3. by interval notation – – for for example, in a function
= −
, the domain of this function can be
written in the form, (-∞, 3) U (3, ∞). This means that the values of the domain can take all real values of x except 3, otherwise the function is undefined. In the succeeding activities, you will learn how to find the domain and the range using different methods. But first let us have another activity that will facilitate the understanding of these methods. Example Find the domain and range of the rational function
= 2 3 = 0 =0
first, we equate the denominator
, therefore
Domain: {x | x ϵ R, x ≠ 0} or simply {x ϵ R | x ≠ 0}, that is all values can take the variable x except 0 because when the denominator denominator bec becomes omes 0, f(x) w will ill be undefine undefinedd (undef). To find the range, we use
=
so that,
= 22 3 3 2 3 = 0 4 4 ≥ 0 = = 2 = 3 2 43 ≥ 0 4 12 ≥ 0 4 ≥ 12
Use Let
,
,
= = 23 3
to get real solutions
Therefore,
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HIGH SCHOOL DEPARTMENT
≥
In summary, D(x) = {x ϵ R | x ≠ 0} and the Range is {y ϵ R | y ≤ 1/3}. Example Find the domain and range of the rational function
2 = 2 2
first, we equate the denominator x + 2 = 0, therefore x = -2 Domain: {x | x ϵ R, x ≠ -2}, that is all values can take the variable x except -2 because the denominator becomes 0 and f(x) will be uundefined. ndefined. The in interval terval notation can also be wri written tten as D (-∞, -2) U (-2, ∞). To find the range, we use
=
so that,
= 2 2
in solving this, you just multiply y and the denominator x + 2 so that it becomes,
2 = 2 = 2 2 22 11 = 22 1 = −+ − 1 = 0 =
Equate
therefore, y ≠ 1, otherwise the denominator is zero.
Range: {y | y ϵ R, R, y ≠ 1}, that is all values can take the variable y except 1 because the denominat denominator or becomes 0 and x will be undefined.
Example Find the domain of the rational function
−− = +−
first, we equate the denominator
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
2 7 4 = 0
,
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HIGH SCHOOL DEPARTMENT by factoring we have, (2x - 1) (x + 4) = 0 therefore x = ½, x = -4 Domain: {x ϵ R | x ≠ -4, 1/2}, that is all values can take the variable x except -4 and 1/2 because the denominator becomes 0 and f(x) will be undefined. The interval notation can also be written as D(-∞, -4) U (-4, ½) U (1/2, ∞).
Example Find the domain and range of the rational function
3 4 = 1 1
first, we equate the denominator x + 1 = 0, therefore x = -1 Domain: {x ϵ R | x ≠ -1}, that is all values can take the variable x except -1 because the denominator becomes 0 and f(x) will be uundefined. ndefined. The in interval terval notation can also be wri written tten as D (-∞, -1) U (-1, ∞). To find the range, we can factor first the numerator.
4 = 11 1
You can cancel both (x + 1) of the numerator and denominator so that what remain is f(x) = (x -4). Then we substitute x = -1 to find y.
= 4 = 1 4 =
Therefore, the Range: {y ϵ R | y ≠ -5}. In interval notation, (-∞, -5) U (-5, ∞).
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HIGH SCHOOL DEPARTMENT Note: Note:
After you read and analyze Lesson 2, you may now proceed to Activity number 2. Activity 2 in General Mathematics Mathematics Lesson 2: Rational Functions Functions
Name: ______________________ __________________________________ ____________
Score: _______ ____________ _____
Grade & Section: ____________ _________________________ _____________
Date: _____________ _____________
Directions: Determine whether the given is a rational function, rational equation, rational inequality or none of these.
+ = 4 − 2. 5 ≥ − − 3. = + 3 4. + − + = 3 5. 3 d, there is no horizontal asymptote.
=
.
A rational function may or may not cross its horizontal asymptote. If the function does not cross the horizontal asymptote y=b, then b is not part of the range of the rational function. f unction. Example Determine the horizontal asymptote of each ea ch rational function. a.
= +
+ GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
b.
= −
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HIGH SCHOOL DEPARTMENT Solutions: a. The degree of the numerator 3x + 8 is less than the degree of the denominator x2 + 1. Therefore, the horizontal asymptote is y = 0.
8 1
b. The degree of the numerator is greater than the degree of the denominator Therefore, there is no horizontal asymptote.
1
.
Aside from vertical and horizontal asymptote, a rational function can have another asymptote called oblique or slant. It occurs when there is no horizontal asymptote or when the degree of the numerator is greater than the degree of the denominator.
Slant / Oblique Asymptote
An oblique asymptote is a line that is neither vertical nor horizontal. It occurs when the numerator of ( ) has a degree that is one higher than the degree of the denominator.
Looking at the graph we can see that there is vertical asymptote and there is no horizontal asymptote. In this case, oblique or slant asymptote occurs. We can determine the oblique / slant asymptote using your knowledge of division of polynomials.
Finding Oblique or Slant Asymptote
To find slant asymptote simply divide the numerator by the denominator by either using long division or synthetic division. The oblique asymptote asymptote is the quotient with the remainder ignored and set equal to y.
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HIGH SCHOOL DEPARTMENT Example Consider the function
ℎ = −+
. Determine the asymptotes. a symptotes.
By looking at the function, h(x) is undefined at x = 1, so the vertical asymptote of h(x) is the line at x = 1. There is no horizontal asymptote because the degree of the numerator is greater than the degree of the denominator.
Remember intercept, substitute 0 for x and solve for y or f(x). To find the y – intercept,
To find the x – intercept, intercept, substitute 0 for y and solve for f or x. intercept of the function. The zero of a rational function is the same as the x – intercept
Problems Involving Rational Functions, Equations, and Inequalities
To be able to solve problems involving rational functions, equations, and inequalities, it is necessary to know the basics of algebra. Solving rational equations and inequalities is very essential in solving word problems. Real-life problems like mixture, work, distance, number, and other related problems might interest you. Example:
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HIGH SCHOOL DEPARTMENT Bamban National High School is preparing for its 25th founding anniversary. The chairperson of the activity allocated ₱90,000.00 from different stakeholders to be divided among various committees of the celebration. Construct a function ( ) which would give the amount of money each of the numbers of committees would receive. If there are six committees, how much would each committee have?
Solution: The function
=
would give the amount of money each of the numbers of committees since
the allocated budget is ₱90,000.00 and it will be divided equally to the number of committees.
6 = 90000 6 = 15000
If there are six committees, then you need to solve for
, thus
Therefore, each committee will receive ₱15,000.00.
Example Barangay Masaya allocated a budget amounting to ₱100,000.00 to provide relief goods for each family in
the barangay due to the Covid-19 pandemic situation. The amount is to be allotted equally among all the families in the barangay. At the same time a philanthropist wants to supplement this budget and he allotted an additional ₱500.00 to be received by each family. Write an equation representing the relationship of the allotted amount per family (y-variable) versus the total number of families (x-variable). How much will be the amount of each relief packs if there are 200 families in the barangay? Solution: The amount to be received by each family is equal to the allotted (₱100,000.00), (₱100,000.00), divided by the number of families plus the amount to be given by the philanthropist. Thus the rational function is described as
=
500. The amount of each relief packs can be computed by finding the value of when = 200,
since there are 200 families in the barangay. Thus,
= 100000 200 500 = 1000
Therefore, the amount of each relief packs to be distributed to each family worth ₱1,000.00.
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HIGH SCHOOL DEPARTMENT Note: Note:
After you read and analyze Lesson 3, you may now proceed to Activity number 3.
Activity 3 in General Mathematics Mathematics Lesson 3: Intercept, Zeroes and and Asymptotes of Rational Rational Functions Functions
Name: ______________________ __________________________________ ____________
Score: _______ ____________ _____
Grade & Section: ____________ _________________________ _____________
Date: _____________ _____________
Directions: Complete the table below by giving the intercepts and zeroes of rational function. Rational Function 1. 2.
3.
= +−
x-intercept
y-intercept
Zeroes of the Function
+ = −+ = −+
Directions: Determine the vertical and horizontal asymptotes of the following rational functions.
1. = + + 2. = +
Vertical Asymptote
Horizontal Asymptote
Directions: In your own words, write the different steps to solve real-life problems involving rational functions, equations, and inequalities. __________________________ _______________ ________________________ __________________________ __________________________ ________________________ ____________________ _________ __________________________ _______________ ________________________ __________________________ __________________________ ________________________ ____________________ _________ __________________________ _______________ ________________________ __________________________ __________________________ _________________________ ____________________ ________ __________________________ _______________ ________________________ __________________________ __________________________ ________________________ ____________________ _________ __________________________ _______________ ________________________ __________________________ __________________________ ________________________ ____________________ _________
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HIGH SCHOOL DEPARTMENT
Inverse Function Learning Objectives: a. determine the inverse of a one-to-one function; b. solve the domain and range of an inverse function; and c. represent an inverse function through its table of values, and graph. Content Standard:
The learners demonstrate understanding of key concepts of inverse functions, exponential functions, and logarithmic functions.
Performance Standard:
The learners are able to apply the concepts of inverse functions, exponential functions, and logarithmic functions to formulate and solve real-life problems with precision and accuracy.
Overview Function is defined as “a relation in which each element of the domain corresponds to exactly one element of the range.” In this module, we will represent
real-life situations using functions, evaluate functions, perform operations on functions, determine the inverse of a function, find the domain and range of inverse function, and graph inverse functions.
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HIGH SCHOOL DEPARTMENT Lesson Discussion: The Inverse of One-to-one Functions Inverse Function Defined
The inverse of a function is a function with domain B and range A given that the original function has domain A and range B.
−
− =
if and only This inverse function of function is denoted by . It is defined by the equation if for any y in range B. Since both are functions, then a function has to be one-to-one for its inverse to be a function at the same time. If it is a many-to-one function, its inverse is one-to-many which is not a function.
=
Find the Inverse of One-to-one Function
Intuitively, the inverse of a function may be known by the principle of “undo”. That is, by considering the inverse of operations performed, the inverse of a function may be computed easily.
Example Given
= 33 8 8
, the inverse of a function may be solved intuitively.
Solution: Steps 1. The last operation performed is subtraction, the inverse operation of which is addition. To x, add 8. 2. The second to the last operation performed is multiplic multiplication, ation, the inverse operation of which is division. Divide x + 8 by 3. to denote that it is the inverse function of 3. Equate it to ( ) = 3 – – 8. 8.
−
In Symbols
3 8 8 3 8 − = 3
However, it is not that easy in some case. In later examples, you will understand what I mean by saying that there is a more general method that may be followed. To find the inverse of a one-to-one function, consider the following:
a. Express the function in the form = ( ); b. Interchange the x and y variables in the equation; c. Solve for y in terms of x.
Example Find the inverse of the rational function
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
ℎ = + −
.
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HIGH SCHOOL DEPARTMENT Solution:
= + −
(change h(x) to y)
(interchange x and y)
− = + 3 = 44 8 8 4 = 3 8 4 4 = 33 8 8 = + − ℎ− = + −
(solve for y, MPE)
(solve for y, by APE)
(solve for y, by factoring)
(solve for y, by MPE)
(the inverse function)
Domain and Range of Inverse Functions
− −
−
The outputs of the function are the inputs to , so the range of is also the domain of . Likewise, because the inputs to are the outputs of , the domain of is the range of . We can visualize the situation.
−
This means that the domain of the inverse is the range of the original function and that the range of the inverse is the domain of the original function.
Original Function X Y
2 6
3 8
5 12
Inverse Function 10 21
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
X y
6 2
8 3
12 5
21 10
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HIGH SCHOOL DEPARTMENT The domain of the original function is (2,3,5,10) and the range is (6,8,12,21). Therefore the domain of the inverse relation will be (6,8,12,21) and the range is (2,3,5,10). ( 2,3,5,10). Properties of an Inverse Function If the
inverse function exists,
− −
1. is a one to one function, f is also one-to-one. 2. Domain of 3. Range of of f.
− − = =
Example Find the domain and range of the inverse function
− = +
Solution: To find the domain and range of an inverse function, go back to the original function and then interchange the domain and range of the original function. The original function is
= 33 2 2
. The original function’s domain is the set of real numbers and the
range is also the set of real numbers. Thus, the domain and range of numbers.
− = +
is the set of all real
Example Find the domain and range of
= 33 1212
and its inverse.
Solution: Let
= 33 1212
Interchange x and y:
= 3 12
Solve for y.
3 = 12 12 = + − = +
− = 33 12 12 − = + } } = ∈ = ∈ − = = ∈ }
Determine the domain and range of and You have
and
Domain
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
.
Range
Domain
− = ∈ }
Range
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HIGH SCHOOL DEPARTMENT Note: Note:
After you read and analyze Lesson 4, you may now proceed to Activity number 4. Activity 4 in General Mathematics Mathematics Lesson 4: Inverse Functions Functions
Name: ______________________ __________________________________ ____________
Score: _______ ____________ _____
Grade & Section: _____________ _________________________ ____________
Date: _____________ _____________
Directions: Intuitively, Intuitively, give the inverse function of each of the following. 1. 2. 3. 4. 5.
== 12 2 12 1 ℎ = = + = = 22 1 1
Directions: Find the inverse of f . Determine the domain and range of each resulting inverse functions. Write your answer inside the box provided. 1.
− =
Solution:
− =
Solution:
Domain Range
2.
= 55 2 2
Domain Range
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HIGH SCHOOL DEPARTMENT
Exponential Functions Learning Objectives: a. distinguish between exponential function, exponential equation, and exponential inequality; b. solve exponential equations and inequalities; and c. represent an exponential function through its table of values, graph, and equation. Content Standard:
The learners demonstrate understanding of key concepts of inverse functions, exponentia exponentiall functions, and logarithmic functions.
Performance Standard:
The learners are able to apply the concepts of inverse functions, exponential functions, and logarithmic functions to formulate and solve real-life problems with precision and accuracy.
Overview An exponential function is defined as “a mathematical function in which an independent variable appears in one of the exponents.” It is almost exclusively used to mean the natural exponential function ex , where e is Euler’s number approximately
equal to 2.718281828…
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HIGH SCHOOL DEPARTMENT Lesson Discussion: Exponential Functions, Equations and Inequalities
An exponential function is a function involving exponential expression showing a relationship between . Examples of which are the independent variable x and dependent variable or and .
10
= 2+ =
On the other hand, an exponential equation is an equation involving exponential expression that can be solved for all x values satisfying the equation. For instance, and .
121 = 11 3 = 9− 64/ > 2 0.9 > 0.81
Lastly, an exponential inequality is an inequality involving exponent exponential ial expression that can be solved for all x values satisfying the inequality. For example, and . Solving Exponential Equations and Exponential Inequalities Exponential Equation
= ≠
≠
One-to-one Property of Exponential Functions states that in , if , then . . This property paves the way in understanding how to solve , then Conversely, if exponential equation.
=
Example
Solve for the value of x in
= 4+ = 64 .
Solution:
4+ = 4 1 1 = 3 1 1 = 3 1 =2
4
Express 64 as
, in order for both sides of the equation to have same bases.
One-to-one Property of Exponential Functions states that if
= = , then
Use Addition Addition Property of Eq Equality uality in order to solv solvee for the valu valuee of x
Combine like terms
Example Solve for the value of x in
3 = 9+
.
Solution:
3 = 9+ 4 = 22 2 2 4 2 = 22 2 2 2 2 = 2
Express 9 as
, in order for both sides of the equation to have same bases.
One-to-one Property of Exponential Functions states that if
3
= = , then
Use Addition Property of Equality in order to solve for the value of x
Combine like terms. Use Multiplication Property of Equality by multiplying both sides of the equation by ½.
=1
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HIGH SCHOOL DEPARTMENT Exponential Inequality
Recall that in an exponential function , but . Now, the key to solving exponential inequality is the fact that if and , then , , Otherwise, if . Let us further make this clearer by considering the next examples.
> 1 > = >0 > ≠ 1
0 < < 1 <
Example Solve for the values of x in
5 > 125+
Solution
5 > 5+ 5 > 3 24 = 5 > 1 > > 3 3 > 2244 < 24 2< 12 > 125+ 5 < 12 ∞,12
Express 125 as
. It is a fact that if
, for both sides of the inequality to have same bases. , then
and
.
Use Addition Property of Equality in order to solve for the value of x.
Combine like terms. Use Multiplication Multiplication Property of Equality by multiplying both sides of the equation by ½.
Hence, the solution to the exponential inequality In symbols, that is, or .
is the set of all real numbers less than -12.
Example Solve for the values of x in
+ ≤ −
.
Solution:
+ ≤ − 2 9 ≤ 33 15 15 = 2 3 ≤ 99 15 ≤ 24 Express
as
, in order for both sides of the inequality to have same bases.
It is a fact that if 0 < < 1 and
< > , then
Use Addition Addition P Property roperty ooff Equality in order to solve for the value of x.
Combine like terms.
+ ≤ − ≤ 24 ∞,24
Thus, the solution to the exponential inequality or equal to 24. In symbols, that is,
or
is the set of all real numbers less than
.
Domain and Range of Exponential Functions
The domain of a function is the set of all allowable values of , commonly known as the independent variable or possible inputs of the function.
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HIGH SCHOOL DEPARTMENT The range of a function is the set of output values commonly known as the dependent variable when all x-values in the domain are evaluated into the function. f unction. This means that you need to find the domain first to describe the range. The Domain and Range
The domain of a function is the set of input values that are used for the independent variable. The range of a function is the set of output values for the dependent variable. For any exponential function, the domain is the set of all real numbers. The range, however, is bounded by the horizontal asymptote of the graph of .
=
Example Find the domain and range of the function
= 3+
.
Solution:
The function is defined for all real numbers. So, the domain of the function is a set of re real al numbers. As extends to approach positive infinity (+∞), the value of the function also extends to +∞, and as extends to approach negative infinity (−∞), the function f unction approaches approaches the x-axis but never touches it. Therefore, the range of the function is a set of real positive numbers greater than 0 or
| ∈ ℝ,ℝ, > 0}.
Thus, the domain and range of the given function is given below and can be written as: Domain
Set Notation Interval Notation
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
∞, | ∈ℝ,ℝ∞∞, }
Range
|∈0,∞ } ℝ ℝ, , > 0} 0 0,∞
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HIGH SCHOOL DEPARTMENT Domain and Range of Exponential Functions
Let
= ∙ ℎ
be an exponential function where ( ) is linear. Then Domain of the function is ℝ
Domain of the function is ℝ
= {ℎ,∞ ℎ,∞ ∞,ℎ ∞,ℎ, >< 0
In cases of exponential functions where ( ) is linear, in which case,
will always be defined for any
value of x. Thus, the domain of an exponential function is the set of real numbers or ℝ. For the range, note that for any a ny values of x. Hence, the range of an exponential function will depend on a and h.
> 0
Example Let
= 4+ 2
. Find the domain and range.
The domain of the function is the set of real numbers, because ( ) = + 1 and it is linear. Also, in the given function you may observe that > 0 ( = 1 = 2) and ℎ = −2, hence the range of the function is equal to (ℎ, +∞).
Set Notation Interval Notation
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
| ∈ℝℝ}∞∞} ∞, Domain
|∈2,ℝ, ℝ,∞∞> 2} 2} Range
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HIGH SCHOOL DEPARTMENT Note: Note:
After you read and analyze Lesson 5, you may now proceed to Activity number 5. Activity 5 in General Mathematics Mathematics Lesson 5: Exponential Functions Functions
Name: ______________________ __________________________________ ____________
Score: _______ ____________ _____
Grade & Section: ____________ _________________________ _____________
Date: _____________ _____________
Directions: Below is a list of exponential expressions. Classify each as to whether it is an exponential function, equation, inequality, inequality, or does not belong to any of these three.
32− ≤ 16+ 36 = 6 6 > ( 1 ) 1 < 1100 = 64 = 2+ 36 100 > 10 = = 4 = 5− 1 + 1 = 2 (2) = (8) 7 = 49 27 < 3 = 5+
Exponential Function
Exponen Exponential tial Equation
Exponential Inequality
None of these
Directions: Solve for the values of x for f or each of the following exponential equatio equations ns and inequalities. 1. 2.
8− = 2 <
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3. 4. 5.
HIGH SCHOOL DEPARTMENT
= 25− 53+ ≥ 27− 4 = 8
Directions: Answer the guide questions to complete the table of domain and range of the following exponential functions. a.
= =55
Is defined at any values of ? __________ What is the minimum value of ( )?_____________ Can you determine the the maximum value of ( )?__________
Domain
Range
Set Notation Interval Notation
b.
= =
Is
defined at any values of ?_________ defined
What is the minimum value of ( )?_____________ Can you determine the the maximum value of ( )?__________
Domain
Range
Set Notation Interval Notation
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
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Community Colleges of the Philippines 301 Mabini Street Quezon District, Cabanatuan City, Nueva Ecija Tel. no. (044) 600-1487 E-mail:
[email protected] [email protected] om
HIGH SCHOOL DEPARTMENT
Logarithmic Functions Learning Objectives: a. distinguish logarithmic function, logarithmic equation, and logarithmic inequality; b. determine the domain and range of a logarithmic function function;; c. solve problems involving logarithmic functions, equations, and inequalities. Content Standard:
The learners demonstrate understanding of key concepts of inverse functions, exponentia exponentiall functions, and logarithmic functions.
Performance Standard:
The learners are able to apply the concepts of inverse functions, exponential functions, and logarithmic functions to formulate and solve real-life problems with precision and accuracy.
Overview An exponential function is defined as “a mathematical function in which an independent variable appears in one of the exponents.” It is almost exclusively used to mean the natural exponential function ex , where e is Euler’s number approximately
equal to 2.718281828…
GENERAL MATHEMATICS GRADE 11 S.Y. 2021-2022
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Community Colleges of the Philippines 301 Mabini Street Quezon District, Cabanatuan City, Nueva Ecija Tel. no. (044) 600-1487 E-mail:
[email protected] [email protected] om
HIGH SCHOOL DEPARTMENT Lesson Discussion: Exponential Equation and Logarithmic Equation
In writing logarithmic equation to exponential function, it can be recalled that the logarithmic function is the inverse of the exponential function and you just need to remember that you are answering the question “To what power must b be raised to obtained the number x?”.
=
=
Example 1.
64 = 3 46 = 64 36 = 6 = 2 2− =
In this example, b=4, y=3 and x=64
2.
In this example, b=36, y= 1/2 and x=6
3.
In this example, b=2, y= -2 and x= ¼
There are exponential equations that are not easy to solve. For instance, the equation
=
cannot be easily solved but for sure, it has a solution. Since
2 < 3 < 2
, therefore,
1 0 = 2 2 33
It is a function of the form , such that and .
=≠1 Definition
Logarithmic Equation
It is an equation involving logarithms. Definition
Logarithmic Inequality
Examples
It is an inequality involving logarithms.
Examples
4 = 3 3 = 2
Examples
< 4 5 5 ≥ 4
Basic Properties and Laws of Logarithm
Let b, x and y be real numbers such that b > 0 and b ≠ 1, the basic properties and laws of logarithms are as follows:
GENERAL MATHEMATICS GRADE 11
S.Y. 2021 2022
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Community Colleges of the Philippines 301 Mabini Street Quezon District, Cabanatuan City, Nueva Ecija Tel. no. (044) 600-1487 E-mail:
[email protected] [email protected] om
HIGH SCHOOL DEPARTMENT Properties of Logarithmic Equations Equations
If b > 0, then the logarithmic function
=
is increasing for all x.
If 0 < b < 1, then t hen the logarithm logarithmic ic function This means that
=
is decreasing for all x.
=
if and only if u v .
Here are some techniques or strategies in solving the logarithmic equation. 1. 2. 3. 4. 5.
Rewriting to exponential form. Using logarithmic properties. Applying the one – to to – one one property of logarithmic functions. The Zero Factor Property: If ab = 0, then a = 0 or b = 0. Take into consideration the domain of logarithm logarithmic ic expression.
Examples Find the value of x in the following. 1.
3 = 22 3
Solution:
3 = 22 3 3 3 = 2222 = 22 3 = 19
Given
One-to-one Property
Addition Property of Equality
Simplify
2.
9 9 8 = 4 99 8 = 4 − = 4 = 3 − 9 = 81 81 8 8 9 = 81 648 648 72 = 648 Solution:
Given
Quotient Law of Logarithm
GENERAL MATHEMATICS GRADE 11
Change into exponential form Multiplication Property of Equality Distributive Property Addition Property of Equality
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Community Colleges of the Philippines 301 Mabini Street Quezon District, Cabanatuan City, Nueva Ecija Tel. no. (044) 600-1487 E-mail:
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=9
HIGH SCHOOL DEPARTMENT Multiplication Property of Equality
Solving Logarithmic Inequalities Inequalities
Remember: If b > 0, then the logarithmic function If 0 < b < 1, then t hen the logarithm logarithmic ic function
>
is increasing for all x.
= =
is decreasing for all x.
This means that implies a > b. Moreover, bear in mind that the domain of the logarithmic function is the set of all positive real numbers. The techniques or strategies in solving logarithmic inequality are the same in solving logarithm logarithmic ic equations.
Example Find all values of x that will satisfy the inequality. 1.
22 1 1 < 3
Solution:
22 1 1 < 3 2 < 22 1 1 8 < 22 1 1 7 < 2 <
Given
Changing into exponential form
Simplify
Addition Property of Equality
Multiplication Property of Equality
2 2 11
Since the domain of logarithmic function is the set of all positive real numbers, the given will be defined if x > -1/2 (2 + 1 > 0 > −1/2). Therefore, the solution set of the inequality is still x > 7/2.
2.
9 < 2
⇒
Solution:
9 > 2 9 > 9 > < 3 < 3
Given
Laws of Logarithm
One-to-one Property
GENERAL MATHEMATICS GRADE 11
Taking square root on both sides.
S.Y. 2021 2022
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Community Colleges of the Philippines 301 Mabini Street Quezon District, Cabanatuan City, Nueva Ecija Tel. no. (044) 600-1487 E-mail:
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HIGH SCHOOL DEPARTMENT Since the domain of the logarithmic function is the set of all positive real numbers, the given be defined if x > 0. Therefore, tthe he solution set of the in inequality equality is 0 < x < 3.
2
will
Domain and Range of Logarithmic Function
The domain of a function is the set of all possible values of the independent variable x. The possible values of the independent variable x are often called inputs. The range of the function are the corresponding values of the dependent variable y. The corresponding values of the dependent variable y are often called outputs. In the case of a logarithmic function, its domain is defined as a set of all positive real numbers while its range is a set of real numbers.
Transformation of the parent function ( ) = either by shift, stretch, compression, or reflection changes c hanges the domain of the parent function. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero. Example 1. Find the domain and range of Solution:
2 4 > 0 2 > 4 2,∞ ∞,∞
Domain: Range:
set up an inequality showing an argument greater than zero
solve for x
write the domain in interval notation
GENERAL MATHEMATICS GRADE 11
S.Y. 2021-2022
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Community Colleges of the Philippines 301 Mabini Street Quezon District, Cabanatuan City, Nueva Ecija Tel. no. (044) 600-1487 E-mail:
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HIGH SCHOOL DEPARTMENT Graph From the graph of the function , it can be seen that the curve is asymptotic at x = 2. Therefore the domain and range are as follows:
2,∞ ∞,∞
Domain: Range:
2. Find the domain and range of . Solution:
Graph:
3 2 > 0
2> −−> 3 > ∞, ∞,∞
Domain:
Range:
GENERAL MATHEMATICS GRADE 11
S.Y. 2021-2022
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Community Colleges of the Philippines 301 Mabini Street Quezon District, Cabanatuan City, Nueva Ecija Tel. no. (044) 600-1487 E-mail:
[email protected] [email protected] om
HIGH SCHOOL DEPARTMENT Note: Note:
After you read and analyze Lesson 5, you may now proceed to Activity number 5. Activity 6 in General Mathematics Mathematics Lesson 6: Logarithmic Functions Functions
Name: ______________________ __________________________________ ____________
Score: _______ ____________ _____
Grade & Section: ____________ _________________________ _____________
Date: _____________ _____________
Directions: Complete the following statements by writing the correct word or words. 1. Logarithm is the inverse of _____________________. 2. The logarithm of a with base b is denoted by ____________ , and is defined as only if____________. 3. In logarithmic form ,the value of cannot be ______________. 4. The value of can be _______________.
=
if and
Directions: Find the value/s of in the following equations/inequalities. 1. 2. 3. 4. 5.
25 = 3 3 3 = 33 2 2 < 2 1 > 2 4 < 5 4
Directions: Graph the following logarithmic functions using an online graphing calculator then find its domain and range. 1.
2.
= ( − 2)
= ( − 1)
Prepared by: Senior High School Teacher
GENERAL MATHEMATICS GRADE 11
S.Y. 2021-2022
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Community Colleges of the Philippines 301 Mabini Street Quezon District, Cabanatuan City, Nueva Ecija Tel. no. (044) 600-1487 E-mail:
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HIGH SCHOOL DEPARTMENT References:
Lesson 1 Queaño (2020). General Mathematics-“Functions.” Mathematics-“Functions.” Published by the Department of Education Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 8-16. Queaño (2020). General Mathematics-“Evaluating Mathematics-“Evaluating Functions.” Published by the the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 40-43. Queaño (2020). General Mathematics-“Operations Mathematics-“Operations on Functions.” Published by the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 6773. Lesson 2 Jolo (2020). General Mathematics-“ Mathematics-“Rational Functions, Equations and Inequalities.” Inequalities.” Published by the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 134-136. Regio (2020). General Mathematics-“ Mathematics-“Solving Rational Equations and Inequalities.” Published by the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 159-170. De Mesa (2020). General Mathematics-“Solving Mathematics- “Solving Rational Equations and Inequalities.” Published by the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 198-199. Quitain (2020). General Mathematics-“ Mathematics-“The Domain and Range of Rational Functions.” Functions .” Published by the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800.
GENERAL MATHEMATICS GRADE 11
S.Y. 2021-2022
Page | 59
Community Colleges of the Philippines 301 Mabini Street Quezon District, Cabanatuan City, Nueva Ecija Tel. no. (044) 600-1487 E-mail:
[email protected] [email protected] om
HIGH SCHOOL DEPARTMENT Pp. 222-227. Lesson 3 Jolo (2020). General Mathematics-“ Mathematics-“Intercepts, Zeroes, and Asymptotes of Rational Functions.” Published by the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 240-261. Vergara (2020). General Mathematics-“Solving Mathematics-“Solving Real-Life Real-Life Problems Involving Rational Functions, Equations, and Inequalities.” Published by the Department of Karangalan
Education Gate 2
Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 283-290. Lesson 4 Gallano (2020). General Mathematics-“The Mathematics-“The Inverse of One-to-OneOne-to-One-Functions.” Functions.” Published Published by the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 336-338. Ibarrola (2020). General Mathematics-“ Mathematics-“Domain and Range of Inverse Functions.” Functions.” Published by the Department of Education Gate 2 Karangalan Village, Barangay San San Isidro Cainta, Rizal 1800. Pp. 382-389. Lesson 5 Mercado (2020). General Mathematics-“ Mathematics-“Exponential Functions, Equations, and Inequalities.” Inequalities.” Published by the Department of of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 463-465. Mercado (2020). General Mathematics-“Solving Mathematics-“Solving Exponential Equations and Inequalities.” Published by
GENERAL MATHEMATICS GRADE 11
S.Y. 2021-2022
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Community Colleges of the Philippines 301 Mabini Street Quezon District, Cabanatuan City, Nueva Ecija Tel. no. (044) 600-1487 E-mail:
[email protected] [email protected] om
HIGH SCHOOL DEPARTMENT the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 483-487. Ibarrola (2020). General MathematicsMathematics-““Domain and Range of Exponential Functions.” Functions .” Published by the Department of of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 525-532. Lesson 6 Asnan (2020). General MathematicsMathematics-“Representing “Representing Real-Life Real-Life Situations Using Logarithmic Functions.” Published by the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Page 603. Asnan (2020). General MathematicsMathematics-“Logarithmic “Logarithmic Functions, Equations and Inequalities.” Published by the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 627-637. Delos Reyes (2020). General Mathematics-“ Mathematics-“Domain and Range of Logarithmic Functions.” Functions.” Published by the Department of Education Gate 2 Karangalan Village, Barangay San Isidro Cainta, Rizal 1800. Pp. 684-685.
GENERAL MATHEMATICS GRADE 11
