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Essential Concepts

Combinations and Compositions of Functions

• We can perform algebraic operations on functions.
• When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.
• The function produced by combining two functions is a composite function.
• The order of function composition must be considered when interpreting the meaning of composite functions.
• A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
• A composite function can be evaluated from a table.
• A composite function can be evaluated from a graph.
• A composite function can be evaluated from a formula.
• Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
• Functions can often be decomposed in more than one way.

Transformations of Functions

• A function can be shifted vertically by adding a constant to the output.
• A function can be shifted horizontally by adding a constant to the input.
• Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts.
• Vertical and horizontal shifts are often combined.
• A vertical reflection reflects a graph about the [latex]x\text{-}[/latex] axis. A graph can be reflected vertically by multiplying the output by –1.
• A horizontal reflection reflects a graph about the [latex]y\text{-}[/latex] axis. A graph can be reflected horizontally by multiplying the input by –1.
• A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph.
• A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly.
• A function presented as an equation can be reflected by applying transformations one at a time.
• Even functions are symmetric about the [latex]y\text{-}[/latex] axis, whereas odd functions are symmetric about the origin.
• Even functions satisfy the condition [latex]f\left(x\right)=f\left(-x\right)[/latex].
• Odd functions satisfy the condition [latex]f\left(x\right)=-f\left(-x\right)[/latex].
• A function can be odd, even, or neither.
• A function can be compressed or stretched vertically by multiplying the output by a constant.
• A function can be compressed or stretched horizontally by multiplying the input by a constant.
• The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order.

Inverse Functions

• If [latex]g\left(x\right)[/latex] is the inverse of [latex]f\left(x\right)[/latex], then [latex]g\left(f\left(x\right)\right)=f\left(g\left(x\right)\right)=x[/latex].
• Each of the toolkit functions, except [latex]y=c[/latex] has an inverse. Some need a restricted domain.
• For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
• A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
• For a tabular function, exchange the input and output rows to obtain the inverse.
• The inverse of a function can be determined at specific points on its graph.
• To find the inverse of a function [latex]y=f\left(x\right)[/latex], switch the variables [latex]x[/latex] and [latex]y[/latex]. Then solve for [latex]y[/latex] as a function of [latex]x[/latex].
• The graph of an inverse function is the reflection of the graph of the original function across the line [latex]y=x[/latex].
• The domain of the inverse function is the range of the original function.
• The range of the inverse function is the domain of the original function.

Key Equations

Composite function	[latex]\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)[/latex]
Vertical shift	[latex]g\left(x\right)=f\left(x\right)+k[/latex] (up for [latex]k>0[/latex] )
Horizontal shift	[latex]g\left(x\right)=f\left(x-h\right)[/latex] (right for [latex]h>0[/latex] )
Vertical reflection	[latex]g\left(x\right)=-f\left(x\right)[/latex]
Horizontal reflection	[latex]g\left(x\right)=f\left(-x\right)[/latex]
Vertical stretch	[latex]g\left(x\right)=af\left(x\right)[/latex] ( [latex]a>0[/latex])
Vertical compression	[latex]g\left(x\right)=af\left(x\right)[/latex] [latex]\left(0
Horizontal stretch	[latex]g\left(x\right)=f\left(bx\right)[/latex] [latex]\left(0
Horizontal compression	[latex]g\left(x\right)=f\left(bx\right)[/latex] ( [latex]b>1[/latex] )

Glossary

composite function

the new function formed by function composition, when the output of one function is used as the input of another

even function

a function whose graph is unchanged by horizontal reflection, [latex]f\left(x\right)=f\left(-x\right)[/latex], and is symmetric about the [latex]y\text{-}[/latex] axis

horizontal compression

a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant [latex]b>1[/latex]

horizontal reflection

a transformation that reflects a function’s graph across the y-axis by multiplying the input by [latex]-1[/latex]

horizontal shift

a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input

horizontal stretch

a transformation that stretches a function’s graph horizontally by multiplying the input by a constant [latex]0

inverse function

for any one-to-one function [latex]f\left(x\right)[/latex], the inverse is a function [latex]{f}^{-1}\left(x\right)[/latex] such that [latex]{f}^{-1}\left(f\left(x\right)\right)=x[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]; this also implies that [latex]f\left({f}^{-1}\left(x\right)\right)=x[/latex] for all [latex]x[/latex] in the domain of [latex]{f}^{-1}[/latex]

odd function

a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\left(x\right)=-f\left(-x\right)[/latex], and is symmetric about the origin

vertical compression

a function transformation that compresses the function’s graph vertically by multiplying the output by a constant [latex]0

vertical reflection

a transformation that reflects a function’s graph across the x-axis by multiplying the output by [latex]-1[/latex]

vertical shift

a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output

vertical stretch

a transformation that stretches a function’s graph vertically by multiplying the output by a constant [latex]a>1[/latex]