Precalculus Combining and Composing Functions Worksheet Solutions (PDF)

Precalculus math worksheet solutions: Combining and Composing Functions – Maths PDF

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This function operations answers for Precalculus covers combining and composing functions (Common Core F-BF.A.1b, F-BF.A.1c), with a free PDF to print.

Check each solution here. Every composite is evaluated from the inside out, with domains reasoned step by step and final answers in bold.

Precalculus · Common Core State Standards: F-BF.A.1b, F-BF.A.1c

Function operations answers: exercises 1 to 10

Exercise 1 – Sum of functions at a point

$f(2)=4$ and $g(2)=5$.

$(f+g)(2)=9$

Exercise 2 – Difference of functions

$3x-2-(x^{2}+1)=-x^{2}+3x-3$.

$-x^{2}+3x-3$

Exercise 3 – Product of functions

$(3x-2)(x^{2}+1)=3x^{3}-2x^{2}+3x-2$.

$3x^{3}-2x^{2}+3x-2$

Exercise 4 – Domain of a quotient

$g(x)=0$ when $x=\pm 2$. Both $f$ and $g$ are defined for all reals.

$(-\infty,-2)\cup(-2,2)\cup(2,\infty)$

Exercise 5 – Evaluate f of g

$g(2)=5$. Then $f(5)=15-2=13$.

Exercise 6 – Evaluate g of f

$f(2)=4$. Then $g(4)=16+1=17$.

Exercise 7 – Composition from a table

Follow each input through g, then f.

$g(1)=0$, $f(0)=2$. $f(0)=2$, $g(2)=2$. $g(3)=1$, $f(1)=5$.

$2,\ 2,\ 5$

Exercise 8 – Composite formula

$f(g(x))=2(x^{2}-1)+3=2x^{2}+1$.

$2x^{2}+1$

Exercise 9 – Composite in the other order

$g(f(x))=(2x+3)^{2}-1=4x^{2}+12x+9-1=4x^{2}+12x+8$.

$4x^{2}+12x+8$

Exercise 10 – Order matters

$f(g(x))=3x+5$. $g(f(x))=3(x+5)=3x+15$.

The results differ because $3x+5\neq 3x+15$

Exercises 11 and up

Exercise 11 – Domain of a composite

The domain starts at 4 and continues right.

Require $g(x)=x-4\ge 0$.

$[4,\infty)$

Exercise 12 – Two composites with a reciprocal

$f(g(x))=\frac{1}{x-3}$, domain $x\neq 3$. $g(f(x))=\frac{1}{x}-3$, domain $x\neq 0$.

$\frac{1}{x-3}$ with $x\neq 3$; $\frac{1}{x}-3$ with $x\neq 0$

Exercise 13 – Domain with a square root inside

Require $x^{2}-9\ge 0$, so $(x-3)(x+3)\ge 0$. The product is nonnegative when $x\le -3$ or $x\ge 3$.

$(-\infty,-3]\cup[3,\infty)$

Exercise 14 – Decompose a power

The inside expression is $5x-1$ and the outer operation is the fourth power.

$g(x)=5x-1,\ f(x)=x^{4}$

Exercise 15 – Decompose a root

Inside: $x^{2}+7$. Outside: the square root.

$g(x)=x^{2}+7,\ f(x)=\sqrt{x}$

Exercise 16 – Three functions

$h(3)=9$, $g(9)=18$, $f(18)=19$.

Exercise 17 – Coupon and discount

Percent first, then coupon.

$s(c(80))=0.8(65)=52$. $c(s(80))=0.8(80)-15=64-15=49$.

$\$52$ and $\$49$; taking the percent off first is cheaper

Exercise 18 – Spreading ripple

At t = 4 s.

$A(r(t))=\pi(3t)^{2}=9\pi t^{2}$. At $t=4$: $9\pi(16)=144\pi\approx 452.4$.

$9\pi t^{2}$; about $452.4\text{ ft}^{2}$

Exercise 19 – Bakery cost model

$C(n(h))=12(8h)+150=96h+150$. For $h=6$: $96(6)+150=726$. For $1206$: $96h=1056$, so $h=11$.

$96h+150$; $\$726$; $11$ hours

Exercise 20 – Verify two functions undo each other

$f(g(x))=\frac{(2x+5)-5}{2}=\frac{2x}{2}=x$. $g(f(x))=2\cdot\frac{x-5}{2}+5=x-5+5=x$.

Both compositions equal $x$

Key facts behind this function operations answers

Key facts

(f+g)(x) = f(x) + g(x); the same pattern holds for difference and product.
(f/g)(x) = f(x)/g(x) requires g(x) not equal to 0.
(f∘g)(x) = f(g(x)): work from the inside out.
In general f(g(x)) is not equal to g(f(x)).
Domain of f(g(x)): x must be in the domain of g, and g(x) in the domain of f.

How to use this function operations answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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