3.1: Composite Functions

Learning Objectives

1. Create a new function by composition of functions.
2. Evaluate composite functions.
3. Find the domain of a composite function.

Composite Functions

If [latex]f[/latex] and [latex]g[/latex] are two functions, the composition of the function [latex]f[/latex] with the function [latex]g[/latex] is written as [latex]f\circ g[/latex].

[latex]\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)[/latex]

We read the left-hand side as “[latex]f[/latex] composed with [latex]g[/latex] at [latex]x[/latex]”) and the right-hand side as “[latex]f[/latex] of [latex]g[/latex] of [latex]x[/latex].”

The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol [latex]\circ[/latex] is called the composition operator.

The domain of composite function must satisfy two things. Informally, any x-value must be

1. In the domain of [latex]g[/latex] (the inside function).
2. In the domain of [latex]f\circ g[/latex] (the simplified expression).

Note: [latex]\left(f\circ g\right)\left(x\right)\neq\left(f\cdot g\right)\left(x\right)[/latex]

Example 3.1-1-1: Find its composite function.

Suppose [latex]f\left(x\right)=x^2[/latex] and [latex]g\left(x\right)=x+2[/latex]

[latex]\left(f\circ g\right)\left(x\right)[/latex]

Key

Example 3.1-1-1: Find its composite function. Suppose [latex]f\left(x\right)=x^2[/latex] and [latex]g\left(x\right)=x+2[/latex]

[latex]\left(f\circ g\right)\left(x\right)[/latex]

[latex]\left(f\circ g\right)\left(x\right)=f\left({\color[rgb]{0.86, 0.0, 0.0}\boldsymbol g}\mathbf{\color[rgb]{0.86, 0.0, 0.0}\left(x\right)}\right)[/latex]

[latex]=f\left({\color[rgb]{0.86, 0.0, 0.0}\boldsymbol x}{\color[rgb]{0.86, 0.0, 0.0}\mathbf+}{\color[rgb]{0.86, 0.0, 0.0}\mathbf2}\right)\;          Use\;x+2\;replaces\;g\left(x\right)[/latex]

[latex]={\color[rgb]{0.0, 0.44, 0.73}\boldsymbol f}\mathbf{\color[rgb]{0.0, 0.44, 0.73}\left({x+2}\right)}[/latex]

[latex]=\mathbf{\left({{\color[rgb]{0.0, 0.44, 0.73}x}{\color[rgb]{0.0, 0.44, 0.73}+}{\color[rgb]{0.0, 0.44, 0.73}2}}\right)}^2\boldsymbol\;          Use\;x+2\;replaces\;x\;in\;function\;f[/latex]

[latex]=x^2+4x+4\boldsymbol\;      Foil\;and\;simplify[/latex]

[latex]\left(f\circ g\right)\left(x\right)=x^2+4x+4[/latex]

Example 3.1-1-2: Find its composite function.

Suppose [latex]f\left(x\right)=x^2[/latex] and [latex]g\left(x\right)=x+2[/latex]

[latex]\left(g\circ f\right)\left(x\right)[/latex]

Key

Example 3.1-1-2: Find its composite function. Suppose [latex]f\left(x\right)=x^2[/latex] and [latex]g\left(x\right)=x+2[/latex]

[latex]\left(g\circ f\right)\left(x\right)[/latex]

[latex]\left(g\circ f\right)\left(x\right)=g\left({\color[rgb]{0.86, 0.0, 0.0}\boldsymbol f}\mathbf{\color[rgb]{0.86, 0.0, 0.0}\left(x\right)}\right)[/latex]

[latex]=g\left({\color[rgb]{0.86, 0.0, 0.0}\boldsymbol x}^{\color[rgb]{0.86, 0.0, 0.0}\mathbf2}\right)        Use\;x^2\;replaces\;f\left(x\right)[/latex]

[latex]={\color[rgb]{0.0, 0.44, 0.73}\boldsymbol g}\mathbf{\color[rgb]{0.0, 0.44, 0.73}\left(x^2\right)}[/latex]

[latex]=\mathbf{\left({\color[rgb]{0.0, 0.44, 0.73}x}^{\color[rgb]{0.0, 0.44, 0.73}2}\right)}+2     Use\;x^2\;replaces\;x\;in\;function\;g[/latex]

[latex]=x^2+2      Simplify[/latex]

[latex]\left(g\circ f\right)\left(x\right)=x^2+2[/latex]

Given a formula for a composite function, evaluate the function.

1. Evaluate the inside function using the input value or variable provided.
2. Use the resulting output as the input to the outside function.

Example 3.1-2-1: Find its composite function.

Suppose [latex]f\left(x\right)=x^2[/latex] and [latex]g\left(x\right)=x+2[/latex]

[latex]\left(f\circ g\right)\left(1\right)[/latex]

Key

Example 3.1-2-1: Find its composite function. Suppose [latex]f\left(x\right)=x^2[/latex] and [latex]g\left(x\right)=x+2[/latex]

[latex]\left(f\circ g\right)\left(1\right)[/latex]

[latex]\left(f\circ g\right)\left(1\right)=f\left({\color[rgb]{0.86, 0.0, 0.0}\boldsymbol g}\mathbf{\color[rgb]{0.86, 0.0, 0.0}\left(1\right)}\right)       Use\;x+2\;replaces\;g\left(x\right)\;and\;x=1,\;thus\;find\;g\left(1\right)[/latex]

[latex]{\color[rgb]{0.86, 0.0, 0.0}\boldsymbol g}\mathbf{\color[rgb]{0.86, 0.0, 0.0}\left(1\right)}{\color[rgb]{0.86, 0.0, 0.0}\mathbf=}{\color[rgb]{0.86, 0.0, 0.0}\mathbf1}{\color[rgb]{0.86, 0.0, 0.0}\mathbf+}{\color[rgb]{0.86, 0.0, 0.0}\mathbf2}{\color[rgb]{0.86, 0.0, 0.0}\mathbf=}{\color[rgb]{0.0, 0.44, 0.73}\mathbf3}[/latex]

[latex]=f\left({\color[rgb]{0.0, 0.44, 0.73}\mathbf3}\right)          Use\;{\color[rgb]{0.0, 0.44, 0.73}\mathbf3}\;replaces\;{\color[rgb]{0.86, 0.0, 0.0}\boldsymbol g}\mathbf{\color[rgb]{0.86, 0.0, 0.0}\left(1\right)}\;in\;function\;f[/latex]

[latex]=\left(3\right)^2          Solve[/latex]

[latex]=9[/latex]

[latex]\left(f\circ g\right)\left(1\right)=9[/latex]

Domain of a Composite Function

The domain of a composite function [latex]f\left(g\left(x\right)\right)[/latex] is the set of those inputs [latex]x[/latex] in the domain of [latex]g[/latex] for which [latex]g\left(x\right)[/latex] is in the domain of [latex]f[/latex].

Given a function composition [latex]f\left(g\left(x\right)\right)[/latex], determine its domain.

1. Find the domain of [latex]g[/latex].
2. Find the domain of [latex]f[/latex].
3. Find those inputs [latex]x[/latex] in the domain of [latex]g[/latex] for which [latex]g\left(x\right)[/latex] is in the domain of [latex]f[/latex]. That is, exclude those inputs [latex]x[/latex] from the domain of [latex]g[/latex] for which [latex]g\left(x\right)[/latex] is not in the domain of [latex]f[/latex]. The resulting set is the domain of [latex]f\circ g[/latex].

Example 3.1-3-1: Finding the domain of the composite function.

[latex]\left(f\circ g\right)\left(x\right)[/latex] where [latex]f\left(x\right)=\sqrt x+2[/latex] and [latex]g\left(x\right)=\sqrt3-x[/latex]

Key

Example 3.1-3-1:Finding the domain of the composite function.

[latex]\left(f\circ g\right)\left(x\right)[/latex] where [latex]f\left(x\right)=\sqrt x+2[/latex] and [latex]g\left(x\right)=\sqrt3-x[/latex]

Rewrite format: [latex]\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)[/latex]

Step 1: find the domain of [latex]g\left(x\right)[/latex].

[latex]g\left(x\right)=3-x,\;x[/latex] can be any number, since it is not under an even root or in the denominator.

Step 2: Find [latex]f\left(g\left(x\right)\right)[/latex].

[latex]f\left(g\left(x\right)\right)=\sqrt{3-x}+2[/latex]

Step 3: Find the domain of [latex]f\left(g\left(x\right)\right)[/latex].

Since [latex]f\left(g\left(x\right)\right)=\sqrt{3-x}+2,\;x[/latex] is under even root, thus restriction applies.

If [latex]\left(A\right)=\sqrt A[/latex], then [latex]A\geq0[/latex]

[latex]3-x\geq0[/latex]

[latex]3\geq x[/latex]

[latex]x\leq3[/latex]

Overall [latex]x\geq0[/latex], and [latex]x\leq3[/latex], thus the real domain of [latex]f\left(g\left(x\right)\right)[/latex] is

[latex][0, 3][/latex]

Note that the domain depends on both the function [latex]f\left(g\left(x\right)\right)[/latex]and the inner function [latex]g\left(x\right)[/latex]. Specifically, we must first ensure that [latex]x[/latex] is in the domain of [latex]g\left(x\right)[/latex] and then that [latex]g\left(x\right)[/latex] lies within the domain of [latex]f[/latex]. In other words, the domain of [latex]f\left(g\left(x\right)\right)[/latex] consists of all values of [latex]x[/latex] such that:

1. [latex]x[/latex] is in the domain of [latex]g[/latex], and
2. [latex]g\left(x\right)[/latex] is in the domain of [latex]f[/latex].