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2.7 Combining Functions

Add, multiply, divide, and compose functions.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Combine functions algebraically and preserve domainsWorked steps · five questions02Read graphs and construct graphical sumsWorked steps · five questions03Evaluate ordered and repeated compositionsWorked steps · five questions04Compose functions from tables and graphsWorked steps · five questions05Find composite domains before simplifyingWorked steps · five questions06Build and decompose two- and three-stage functionsWorked steps · five questions07Model geometric and motion chainsWorked steps · five questions08Model revenue, profit, and ordered discountsWorked steps · five questions09Analyze linear composition and recover unknown functionsWorked steps · five questions10Justify iteration and parity of compositionsWorked steps · five questions
Lesson brief and rules

2.7 Combining Functions

(f(f ∘ g)(x)=f(g(x))g)(x) = f(g(x)); evaluate the inner function first. For fg\frac{f}{g}, exclude zeros of gg.

Remember
Composition order can change the answer.
Worked example 11
01 / For f(x)=f(x)=x(x+2)x \left(x + 2\right) and g(x)=g(x)=x2x^{2}, find f+g,f−gf+g, f-g, fg, and fg\dfrac{f}{g}, with all four domains. Preserve any excluded input after simplifying.
02 / Add and subtract like terms; multiply the two original expressions to obtain f+gf+g: 2x(x+1)2 x \left(x + 1\right); f−gf-g: 2x2 x; fg: x3(x+2)x^{3} \left(x + 2\right).
03 / Divide the original expressions and simplify to x+2x\dfrac{x + 2}{x}.
04 / The original denominator vanishes at 00. All real inputs for sum, difference and product; quotient excludes 00.
Worked example 2 · Combine functions algebraically and preserve domains

For f(x)=f(x)=x(x+3)x \left(x + 3\right) and g(x)=g(x)=x+3x + 3, find f+g,f−gf+g, f-g, fg, and fg\dfrac{f}{g}, with all four domains. Preserve any excluded input after simplifying.

Add and subtract like terms; multiply the two original expressions to obtain f+gf+g: (x+1)(x+3)\left(x + 1\right) \left(x + 3\right); f−gf-g: (x−1)(x+3)\left(x - 1\right) \left(x + 3\right); fg: x(x+3)2x \left(x + 3\right)^{2}.
Divide the original expressions and simplify to xx.
The original denominator vanishes at −3-3. All real inputs for sum, difference and product; quotient excludes −3-3.

Read the graphs

Read graphs and construct graphical sums

Use the graph, not an assumed formula, to find the sum of ff and gg at x=−2x=-2.
Function graphPiecewise-linear f and g; filled endpoints are included and no graph exists beyond them.-3-2-1012345-2-11234567xyfg
Piecewise-linear f and g; filled endpoints are included and no graph exists beyond them.

Read f(−2)=2f(-2)=2 and g(−2)=3g(-2)=3.
Apply the sum to these ordinates, obtaining 55.

Answer: 55

Compose functions from tables and graphs

Follow the two stages on the graph to evaluate f(g(1))f(g(1)).
Function graphBoth functions have domain [0,4]; values at integer inputs are marked.-1012345-112345xyfg
Both functions have domain [0,4]; values at integer inputs are marked.

The inner ordinate is g(1)=0g(1)=0.
Move to input 00 on ff and read output 11.

Answer: 11

15 questions with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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