Domain and Range Cheat Sheet

Harold’s Parent Functions “Cheat Sheet” 28 December 2015

Function Name

Parent Function

Graph

Characteristics

Algebra

Constant

𝑓(𝑥) = 𝑐

Linear or Identity

𝑓(𝑥) = 𝑥

Quadratic or Square

𝑓(𝑥) = 𝑥 2

Square Root

𝑓(𝑥) = √𝑥

Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor

Domain: (−∞, ∞) Range: [c, c] Inverse Function: Undefined (asymptote) Restrictions: c is a real number Odd/Even: Even General Form: 𝐴𝑦 + 𝐵 = 0 Domain: (−∞, ∞) Range: (−∞, ∞) Inverse Function: 𝑔(𝑥) = 𝑥 Restrictions: m ≠ 0 Odd/Even: Odd General Forms: 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 𝑦 = 𝑚𝑥 + 𝑏 𝑦 − 𝑦0 = 𝑚(𝑥 − 𝑥0 ) Domain: (−∞, ∞) Range: [0, ∞) Inverse Function: 𝑔(𝑥) = √𝑥 Restrictions: None Odd/Even: Even General Form: 𝐴𝑥 2 + 𝐵𝑦 + 𝐶𝑥 + 𝐷 = 0 Domain: [0, ∞) Range: [0, ∞) Inverse Function: 𝑔(𝑥) = x 2 Restrictions: 𝑥 ≥ 0 Odd/Even: Neither General Form: 𝑓(𝑥) = 𝑎√𝑏(𝑥 − ℎ) + 𝑘

1

Function Name

Parent Function

Absolute Value

𝑓(𝑥) = |𝑥|

Cubic

𝑓(𝑥) = 𝑥 3

Cube Root

𝑓(𝑥) = √𝑥

Exponential

𝑓(𝑥) = 10𝑥 𝑜𝑟 𝑓(𝑥) = 𝑒 𝑥

Logarithmic

𝑓(𝑥) = log 𝑥 𝑜𝑟 𝑓(𝑥) = ln 𝑥

Graph

3

Copyright © 2011-2015 by Harold A. Toomey, WyzAnt Tutor

Characteristics Domain: (−∞, ∞) Range: [0, ∞) Inverse Function: 𝑓(𝑥) = 𝑥 𝑓𝑜𝑟 𝑥 ≥ 0 Restrictions: 𝑥, 𝑖𝑓 𝑥 ≥ 0 𝑓(𝑥) = { −𝑥, 𝑖𝑓 𝑥 < 0 Odd/Even: Even General Form: 𝑓(𝑥) = 𝑎|𝑏(𝑥 − ℎ)| + 𝑘 Domain: (−∞, ∞) Range: (−∞, ∞) Inverse Function: 3 𝑔(𝑥) = √𝑥 Restrictions: None Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎(𝑏(𝑥 − ℎ))3 + 𝑘 Domain: (−∞, ∞) Range: (−∞, ∞) Inverse Function: 𝑔(𝑥) = 𝑥 3 Restrictions: None Odd/Even: Odd General Form: 3 𝑓(𝑥) = 𝑎 √𝑏(𝑥 − ℎ) + 𝑘 Domain: (−∞, ∞) Range: (0, ∞) Inverse Function: 𝑔(𝑥) = log 𝑥 𝑜𝑟 𝑔(𝑥) = ln 𝑥 Restrictions: None, x can be imaginary Odd/Even: Neither General Form: 𝑓(𝑥) = 𝑎 10(𝑏(𝑥−ℎ)) + 𝑘 Domain: (0, ∞) Range: (−∞, ∞) Inverse Function: 𝑔(𝑥) = 10𝑥 𝑜𝑟 𝑔(𝑥) = 𝑒 𝑥 Restrictions: x > 0 Odd/Even: Neither General Form: 𝑓(𝑥) = 𝑎 log(𝑏(𝑥 − ℎ)) + 𝑘 2

Function Name

Parent Function

Graph

Characteristics Domain: (−∞, 0) ∪ (0, ∞) Range: (−∞, 0) ∪ (0, ∞) Inverse Function:

Reciprocal or Rational

𝑔(𝑥) = 𝑓(𝑥) =

1 𝑥

Restrictions: x ≠ 0 Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 [

Greatest Integer or Floor

𝑓(𝑥) = [𝑥]

Inverse Functions

𝐼𝑓 𝑓(𝑥) = 𝑦, 𝑡ℎ𝑒𝑛 −1 (𝑦) 𝑓 = 𝑓 −1 (𝑓(𝑥)) =𝑥

1 𝑥

𝑏 ]+𝑘 (𝑥 − ℎ)

Domain: (−∞, ∞) Range: (−∞, ∞) whole numbers only Inverse Function: Undefined (asymptotic) Restrictions: Real numbers only Odd/Even: Neither General Form: 𝑓(𝑥) = 𝑎[𝑏(𝑥 − ℎ)] + 𝑘 Domain of x Domain of y Range of y Range of x Inverse Function: By definition Restrictions: None Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑓(𝑏(𝑥 − ℎ)) + 𝑘

Conic Sections

Circle

𝑥2 + 𝑦2 = 𝑟2

Domain: [−𝑟 + ℎ, 𝑟 + ℎ] Range: [−𝑟 + 𝑘, 𝑟 + 𝑘] Inverse Function: Same as parent Restrictions: None Odd/Even: Both Focus : (ℎ, 𝑘) General Forms: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝐶 𝑎𝑛𝑑 𝐵 = 0

Copyright © 2011-2015 by Harold A. Toomey, WyzAnt Tutor

3

Function Name

Ellipse

Parent Function

Graph

𝑥2 𝑦2 + =1 𝑎2 𝑏 2

Characteristics Domain: [−𝑎 + ℎ, 𝑎 + ℎ] Range: [−𝑏 + 𝑘, 𝑏 + 𝑘] Inverse Function: 𝑥2 𝑦2 + =1 𝑏 2 𝑎2 Restrictions: None Odd/Even: Both Foci : 𝑐 2 = 𝑎2 − 𝑏 2 General Forms: (𝑥 − ℎ)2 (𝑦 − 𝑘)2 + =1 𝑎2 𝑏2 𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 where 𝐵2 − 4𝐴𝐶 < 0

Parabola

𝑦 = 𝑎𝑥 2

Domain: (−∞, ∞) Range: [𝑘, ∞) or (−∞, 𝑘] Inverse Function: 𝑔(𝑥) = √𝑥 Restrictions: None Odd/Even: Even Vertex : (ℎ, 𝑘) Focus : (ℎ, 𝑘 + 𝑝) General Forms: (𝑥 − ℎ)2 = 4𝑝(𝑦 − 𝑘) 𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 where 𝐵2 − 4𝐴𝐶 = 0

Hyperbola

𝑥2 𝑦2 − =1 𝑎2 𝑏 2

Domain: (−∞, -a+h] ∪ [a+h, ∞) Range: (−∞, ∞) Inverse Function: 𝑦2 𝑥2 − =1 𝑎2 𝑏 2 Restrictions: Domain is restricted Odd/Even: Both Foci : 𝑐 2 = 𝑎2 + 𝑏 2 General Forms: (𝑥 − ℎ)2 (𝑦 − 𝑘)2 − =1 𝑎2 𝑏2 𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 where 𝐵2 − 4𝐴𝐶 > 0

Copyright © 2011-2015 by Harold A. Toomey, WyzAnt Tutor

4

Function Name

Parent Function

Graph

Characteristics

Trigonometry

Sine

𝑓(𝑥) = 𝑠𝑖𝑛 𝑥

Cosine

𝑓(𝑥) = 𝑐𝑜𝑠 𝑥

Domain: (−∞, ∞) Range: [−1, 1] Inverse Function: 𝑔(𝑥) = 𝑠𝑖𝑛−1 𝑥 Restrictions: None Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑠𝑖𝑛 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, ∞) Range: [−1, 1] Inverse Function: 𝑔(𝑥) = 𝑐𝑜𝑠 −1 𝑥 Restrictions: None Odd/Even: Even General Form: 𝑓(𝑥) = 𝑎 𝑐𝑜𝑠 (𝑏(𝑥 − ℎ)) + 𝑘 𝜋

𝑓(𝑥) = 𝑡𝑎𝑛 𝑥 Tangent =

𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠 𝑥

Domain: (−∞, ∞) except for 𝑥 = 2 ± 𝑛𝜋 Range: (−∞, ∞) Inverse Function: 𝑔(𝑥) = 𝑡𝑎𝑛−1 𝑥 𝜋 Restrictions: Asymptotes at 𝑥 = 2 ± 𝑛𝜋 Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑡𝑎𝑛 (𝑏(𝑥 − ℎ)) + 𝑘 𝜋

𝑓(𝑥) = sec 𝑥 Secant =

1 𝑐𝑜𝑠 𝑥

𝑓(𝑥) = 𝑐𝑠𝑐 𝑥 Cosecant =

1 𝑠𝑖𝑛 𝑥

𝑓(𝑥) = 𝑐𝑜𝑡 𝑥 Cotangent =

1 𝑡𝑎𝑛 𝑥

Copyright © 2011-2015 by Harold A. Toomey, WyzAnt Tutor

Domain: (−∞, ∞) except for 𝑥 = 2 ± 𝑛𝜋 Range: (−∞,−1] ∪ [1, ∞) Inverse Function: 𝑔(𝑥) = 𝑠𝑒𝑐 −1 𝑥 Restrictions: Range is bounded Odd/Even: Even General Form: 𝑓(𝑥) = 𝑎 𝑠𝑒𝑐 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, ∞) except for 𝑥 = ±𝑛𝜋 Range: (−∞, -1] ∪ [1, ∞) Inverse Function: 𝑔(𝑥) = 𝑐𝑠𝑐 −1 𝑥 Restrictions: Range is bounded Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑐𝑠𝑐 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, ∞) except for 𝑥 = ±𝑛𝜋 Range: (−∞, ∞) Inverse Function: 𝑔(𝑥) = 𝑐𝑜𝑡 −1 𝑥 Restrictions: Asymptotes at x = ±𝑛𝜋 Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑐𝑜𝑡 (𝑏(𝑥 − ℎ)) + 𝑘 5

Function Name

Parent Function

Graph

Characteristics Domain: [−1, 1] −𝜋 𝜋

Arcsine

𝑓(𝑥) = 𝑠𝑖𝑛−1 𝑥

Arccosine

𝑓(𝑥) = 𝑐𝑜𝑠 −1 𝑥

Range: [ 2 , 2 ] or Quadrants I & IV Inverse Function: 𝑔(𝑥) = 𝑠𝑖𝑛 𝑥 Restrictions: Range & Domain are bounded Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑠𝑖𝑛−1 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: [−1, 1] Range: [0, 𝜋] or Quadrants I & II Inverse Function: 𝑔(𝑥) = 𝑐𝑜𝑠 𝑥 Restrictions: Range & Domain are bounded Odd/Even: None General Form: 𝑓(𝑥) = 𝑎 𝑐𝑜𝑠 −1 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, ∞) −𝜋 𝜋

Arctangent

𝑓(𝑥) = 𝑡𝑎𝑛−1 𝑥

Arcsecant

𝑓(𝑥) = 𝑠𝑒𝑐 −1 𝑥

Arccosecant

𝑓(𝑥) = 𝑐𝑠𝑐 −1 𝑥

Arccotangent

𝑓(𝑥) = 𝑐𝑜𝑡 −1 𝑥

Copyright © 2011-2015 by Harold A. Toomey, WyzAnt Tutor

Range: ( , ) or Quadrants I & IV 2 2 Inverse Function: 𝑔(𝑥) = 𝑡𝑎𝑛 𝑥 Restrictions: Range is bounded Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑡𝑎𝑛−1 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞,−1] ∪ [1, ∞) 𝜋 𝜋 Range: [0, ) ∪ ( , 𝜋] or Quadrants I & II 2 2 Inverse Function: 𝑔(𝑥) = 𝑠𝑒𝑐 𝑥 Restrictions: Range & Domain are bounded Odd/Even: Neither General Form: 𝑓(𝑥) = 𝑎 𝑠𝑒𝑐 −1 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞,−1] ∪ [1, ∞) 𝜋 𝜋 Range: [ − 2 , 0) ∪ (0, 2 ] or Quadrants I & IV Inverse Function: 𝑔(𝑥) = 𝑐𝑠𝑐 𝑥 Restrictions: Range & Domain are bounded Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑐𝑠𝑐 −1 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, ∞) Range: (0, 𝜋) or Quadrants I & II Inverse Function: 𝑔(𝑥) = 𝑐𝑜𝑡 𝑥 Restrictions: Range is bounded Odd/Even: Neither General Form: 𝑓(𝑥) = 𝑎 𝑐𝑜𝑡 −1 (𝑏(𝑥 − ℎ)) +𝑘

6

Function Name

Parent Function

Graph

Characteristics

Hyperbolics

𝑓(𝑥) = sinh 𝑥 Hyperbolic Sine =

𝑒 𝑥 − 𝑒 −𝑥 2

𝑓(𝑥) = 𝑐𝑜𝑠ℎ 𝑥 Hyperbolic Cosine

𝑒 𝑥 + 𝑒 −𝑥 = 2

𝑓(𝑥) = 𝑡𝑎𝑛ℎ 𝑥 Hyperbolic Tangent

=

𝑒 2𝑥 − 1 𝑒 2𝑥 + 1

𝑓(𝑥) = sech 𝑥 Hyperbolic Secant

=

1 𝑐𝑜𝑠ℎ 𝑥

𝑓(𝑥) = 𝑐𝑠𝑐ℎ 𝑥 Hyperbolic Cosecant

=

1 𝑠𝑖𝑛ℎ 𝑥

𝑓(𝑥) = 𝑐𝑜𝑡ℎ 𝑥 Hyperbolic Cotangent

=

𝑒 2𝑥 + 1 𝑒 2𝑥 − 1

Copyright © 2011-2015 by Harold A. Toomey, WyzAnt Tutor

Domain: (−∞, ∞) Range: (−∞, ∞) Inverse Function: 𝑔(𝑥) = 𝑠𝑖𝑛ℎ−1 𝑥 Restrictions: None Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑠𝑖𝑛ℎ (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, ∞) Range: [1, ∞) Inverse Function: 𝑔(𝑥) = 𝑐𝑜𝑠ℎ−1 𝑥 Restrictions: None Odd/Even: Even General Form: 𝑓(𝑥) = 𝑎 𝑐𝑜𝑠ℎ (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, ∞) Range: (−1, 1) Inverse Function: 𝑔(𝑥) = 𝑡𝑎𝑛ℎ−1 𝑥 Restrictions: Asymptotes at 𝑦 = ±1 Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑡𝑎𝑛ℎ (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, ∞) Range: (0, 1] Inverse Function: 𝑔(𝑥) = 𝑠𝑒𝑐ℎ−1 𝑥 Restrictions: Asymptote at 𝑦 = 0 Odd/Even: Even General Form: 𝑓(𝑥) = 𝑎 𝑠𝑒𝑐ℎ (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, 0) ∪ (0, ∞) Range: (−∞, 0] ∪ [0, ∞) Inverse Function: 𝑔(𝑥) = 𝑐𝑠𝑐ℎ−1 𝑥 Restrictions: Asymptotes at 𝑥 = 0, 𝑦 = 0 Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑐𝑠𝑐ℎ (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, 0) ∪ (0, ∞) Range: (−∞, 1) ∪ (1, ∞) Inverse Function: 𝑔(𝑥) = 𝑐𝑜𝑡ℎ−1 𝑥 Restrictions: Asymptotes at 𝑥 = 0, 𝑦 = ±1 Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑐𝑜𝑡ℎ (𝑏(𝑥 − ℎ)) + 𝑘 7

Function Name

Hyperbolic Arcsine

Hyperbolic Arccosine

Parent Function

Graph

𝑓(𝑥) = 𝑠𝑖𝑛ℎ−1 𝑥 = 𝑙𝑛(𝑥 + √𝑥 2 + 1)

𝑓(𝑥) = 𝑐𝑜𝑠ℎ−1 𝑥 = 𝑙𝑛(𝑥 + √𝑥 2 − 1)

𝑓(𝑥) = 𝑡𝑎𝑛ℎ−1 𝑥 Hyperbolic Arctangent

1 1+𝑥 = 𝑙𝑛 ( ) 2 1−𝑥

𝑓(𝑥) = 𝑠𝑒𝑐ℎ−1 𝑥 Hyperbolic Arcsecant

1 1 = 𝑙𝑛 ( + √ 2 − 1) 𝑥 𝑥

𝑓(𝑥) = 𝑐𝑠𝑐ℎ−1 𝑥 Hyperbolic Arccosecant

1 1 = 𝑙𝑛 ( + √ 2 + 1) 𝑥 𝑥

𝑓(𝑥) = 𝑐𝑜𝑡ℎ−1 𝑥 Hyperbolic Arccotangent

1 𝑥+1 = 𝑙𝑛 ( ) 2 𝑥−1

Copyright © 2011-2015 by Harold A. Toomey, WyzAnt Tutor

Characteristics Domain: (−∞, ∞) Range: (−∞, ∞) Inverse Function: 𝑔(𝑥) = 𝑠𝑖𝑛ℎ 𝑥 Restrictions: None Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑠𝑖𝑛ℎ−1 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: [1, ∞) Range: [0, ∞) Inverse Function: 𝑔(𝑥) = 𝑐𝑜𝑠ℎ 𝑥 Restrictions: 𝑦 ≥ 0 Odd/Even: Neither General Form: 𝑓(𝑥) = 𝑎 𝑐𝑜𝑠ℎ−1 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−1, 1) Range: (−∞, ∞) Inverse Function: 𝑔(𝑥) = 𝑡𝑎𝑛ℎ 𝑥 Restrictions: Asymptotes at 𝑥 = ±1 Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑡𝑎𝑛ℎ−1 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (0, 1] Range: [0, ∞) Inverse Function: 𝑔(𝑥) = 𝑠𝑒𝑐ℎ 𝑥 Restrictions: Odd/Even: Neither General Form: 𝑓(𝑥) = 𝑎 𝑠𝑒𝑐ℎ−1 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: (−∞, 0) ∪ (0, ∞) Range: (−∞, 0] ∪ [0, ∞) Inverse Function: 𝑔(𝑥) = 𝑐𝑠𝑐ℎ 𝑥 Restrictions: Asymptotes at 𝑥 = 0, 𝑦 = 0 Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑐𝑠𝑐ℎ−1 (𝑏(𝑥 − ℎ)) + 𝑘 Domain: [−∞, −1) ∪ (1, ∞] Range: (−∞, 0) ∪ (0, ∞) Inverse Function: 𝑔(𝑥) = 𝑐𝑜𝑡ℎ 𝑥 Restrictions: Asymptotes at 𝑥 = 0, 𝑦 = ±1 Odd/Even: Odd General Form: 𝑓(𝑥) = 𝑎 𝑐𝑜𝑡ℎ−1 (𝑏(𝑥 − ℎ)) +𝑘 8

Graphing Tips All Functions The Six Function “Levers”

y = a f (b (x - h)) + k

Graphing Tips

1) Move up/down ↕

k

(Vertical translation)

“+” Moves it up

2) Move left/right ↔

h

(Phase shift)

“+“ Moves it right

3) Stretch up/down ↕

a

(Amplitude)

Larger stretches it taller or makes it grow faster

4) Stretch left/right ↔

b

(Frequency ⦁ 2π)

Larger stretches it wider

5) Flip about x-axis

a → –a

6) Flip about y-axis

b → –b

𝑓(𝑥) → – 𝑓(𝑥) If 𝑓(𝑥) =– 𝑓(−𝑥) then odd function 𝑓(𝑥) → 𝑓(−𝑥) If 𝑓(𝑥) = 𝑓(−𝑥) then even function

Trigonometric Functions The Six Trig “Levers”

y = a sin (b (x - h)) + k

Graphing Tips (max + min) 2

Notes If 𝑘 = 𝑓(𝑥) then x-axis is replaced by 𝑓(𝑥)-axis

1) Move up/down ↕

k

(Vertical translation)

2) Move left/right ↔

h

(Phase shift)

3) Stretch up/down ↕

a

(Amplitude)

4) Stretch left/right ↔

b

(Frequency ⦁ 2π)

5) Flip about x-axis

a → –a

𝑓(𝑥) → −𝑓(−𝑥)

Odd Function: 𝑠𝑖𝑛 (𝑥) = −𝑠𝑖𝑛 (−𝑥)

6) Flip about y-axis

b → –b

𝑓(𝑥) → 𝑓(−𝑥)

Even Function: 𝑐𝑜𝑠 (𝑥) = 𝑐𝑜𝑠 (−𝑥)

k=

‘+‘ shifts right (max – min) 2 2π 1 T= = |b| ƒ

a=

Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor

𝑠𝑖𝑛 (𝑥) = 𝑐𝑜𝑠 (𝑥 − 𝜋/2) a is NOT peak-to-peak on y-axis T = peak-to-peak on θ-axis 𝜋 𝑇 = 𝑏 for 𝑡𝑎𝑛 (𝑏𝑥)

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