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The Differences between a relation and a function Relation: A relation is simply a set of ordered pairs. The first elements in the ordered pairs (the x-values), form the domain. The second elements in the ordered pairs (the y-values), form the range. Only the elements "used" by the relation constitute the range. This mapping shows a relation from set A into set B. This relation consists of the ordered pairs (1,2), (3,2), (5,7), and (9,8). • The domain is the set {1, 3, 5, 9}. • The range is the set {2, 7, 8}. (Notice that 3, 5 and 6 are not part of the range.) • The range is the dependent variable.

The following are examples of relations. Notice that a vertical line may intersect a relation in more than one location.

This set of 5 points is a relation. {(1,2), (2, 4), (3, 5), (2, 6), (1, -3)} Notice that vertical lines may intersect more than one point at a time.

This parabola is also a relation. Notice that a vertical line can intersect this graph twice.

If we impose the following rule on a relation, it becomes a function.

Function: A function is a set of ordered pairs in which each x-element has only ONE y-element associated with it.

The relations shown above are NOT functions because certain x-elements are paired with more than one unique y-element. The first relation shown above can be altered to become a function by removing the ordered pairs where the x-coordinate is repeated. It will not matter which "repeat" is removed.

function: {(1,2), (2,4), (3,5)}

The graph at the right shows that a vertical line now intersects only ONE point in our new function.

Vertical line test:

each vertical line drawn through the graph will intersect a function in only one location.

Five Representations

Relation: any set of ordered pairs ���������Examples� A = {(2,5), (-7,4.8), (4,12), (2,7)}�� B = {(9,0), (4,5), (8.1,0.23), (4,5)}� C = {(1, red), (2, red)}

Function: a set of ordered pairs that maps each x-value to exactly one y-value ���������In the examples above, B is a function while A is not. Can you explain why this is true?

Typically, we use five representations to describe functions: 1) a set/listing of ordered pairs, 2) a graph, 3) a table of values, 4) an arrow diagram, and 5) a rule/formula.

F = {(3,4.5), (-9,7), (0.8,-12)}� domain D = {3, -9, 0.8} and range R = {4.5, -12, 7}

The graph would be the set of three points mapped onto the Cartesian plane in the usual way. A graph is a picture of the function. Recall that the vertical line test (VLT) can be used to test a graph to determine if it is a function. The criteria that must be met is that the vertical line pass through only one point on the graph at a time.

The table of values would be as follows:����������

x

y

3

4.5

-9

7

0.8

-12

The table tells us that F(-9) = 7 or the value of the function F at -9 is 7.

The arrow diagram is another representation of the function that shows how the domain (x) maps into the range (y).

x

y

3���� �� �

4.5

-9�� �

7

0.8� �

-12

Notice that in an arrow diagram for a function, each x-value has exactly one arrow coming out of it to the same y-value.

The last representation is the rule or formula. Suppose function g(x) = 2x + 5. The rule/formula for g is 2xz + 5 because this determines what happens to x to give the related y.

Kinds of Functions

1.� linear functions:

f(x) = mx + b (highest power on x is 1)

2.� quadratic functions: f(x) = ax2 + bx + c (highest power on x is 2) 3.� exponential functions: f(x) = a(b)x (x is the power in the function) 4.� logarithmic functions: f(x) = a(ln x) + b (x is part of a log/ln function)

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The following are examples of relations. Notice that a vertical line may intersect a relation in more than one location.

This set of 5 points is a relation. {(1,2), (2, 4), (3, 5), (2, 6), (1, -3)} Notice that vertical lines may intersect more than one point at a time.

This parabola is also a relation. Notice that a vertical line can intersect this graph twice.

If we impose the following rule on a relation, it becomes a function.

Function: A function is a set of ordered pairs in which each x-element has only ONE y-element associated with it.

The relations shown above are NOT functions because certain x-elements are paired with more than one unique y-element. The first relation shown above can be altered to become a function by removing the ordered pairs where the x-coordinate is repeated. It will not matter which "repeat" is removed.

function: {(1,2), (2,4), (3,5)}

The graph at the right shows that a vertical line now intersects only ONE point in our new function.

Vertical line test:

each vertical line drawn through the graph will intersect a function in only one location.

Five Representations

Relation: any set of ordered pairs ���������Examples� A = {(2,5), (-7,4.8), (4,12), (2,7)}�� B = {(9,0), (4,5), (8.1,0.23), (4,5)}� C = {(1, red), (2, red)}

Function: a set of ordered pairs that maps each x-value to exactly one y-value ���������In the examples above, B is a function while A is not. Can you explain why this is true?

Typically, we use five representations to describe functions: 1) a set/listing of ordered pairs, 2) a graph, 3) a table of values, 4) an arrow diagram, and 5) a rule/formula.

F = {(3,4.5), (-9,7), (0.8,-12)}� domain D = {3, -9, 0.8} and range R = {4.5, -12, 7}

The graph would be the set of three points mapped onto the Cartesian plane in the usual way. A graph is a picture of the function. Recall that the vertical line test (VLT) can be used to test a graph to determine if it is a function. The criteria that must be met is that the vertical line pass through only one point on the graph at a time.

The table of values would be as follows:����������

x

y

3

4.5

-9

7

0.8

-12

The table tells us that F(-9) = 7 or the value of the function F at -9 is 7.

The arrow diagram is another representation of the function that shows how the domain (x) maps into the range (y).

x

y

3���� �� �

4.5

-9�� �

7

0.8� �

-12

Notice that in an arrow diagram for a function, each x-value has exactly one arrow coming out of it to the same y-value.

The last representation is the rule or formula. Suppose function g(x) = 2x + 5. The rule/formula for g is 2xz + 5 because this determines what happens to x to give the related y.

Kinds of Functions

1.� linear functions:

f(x) = mx + b (highest power on x is 1)

2.� quadratic functions: f(x) = ax2 + bx + c (highest power on x is 2) 3.� exponential functions: f(x) = a(b)x (x is the power in the function) 4.� logarithmic functions: f(x) = a(ln x) + b (x is part of a log/ln function)

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