Precalculus Combining and Composing Functions Worksheet (PDF)

Precalculus math worksheet: Combining and Composing Functions – Maths PDF

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

These 20 problems practice combining and composing functions (Common Core F-BF.A.1b and F-BF.A.1c). You will evaluate sums, products and composites, find domains, decompose functions and solve model problems.

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

Function operations worksheet: exercises 1 to 10

Exercise 1 – Sum of functions at a point

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

Exercise 2 – Difference of functions

For the same $f$ and $g$, write $(f-g)(x)$ in simplest form.

Exercise 3 – Product of functions

For the same $f$ and $g$, write $(fg)(x)$ in standard form.

Exercise 4 – Domain of a quotient

Let $f(x)=x+1$ and $g(x)=x^{2}-4$. Find the domain of $\left(\frac{f}{g}\right)(x)$.

Exercise 5 – Evaluate f of g

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

Exercise 6 – Evaluate g of f

With the same functions, find $g(f(2))$.

Exercise 7 – Composition from a table

Table for f and g.

Use the table to find $f(g(1))$, $g(f(0))$ and $f(g(3))$.

Exercise 8 – Composite formula

Let $f(x)=2x+3$ and $g(x)=x^{2}-1$. Write $(f\circ g)(x)$ in simplest form.

Exercise 9 – Composite in the other order

For the same $f$ and $g$, write $(g\circ f)(x)$ in standard form.

Exercise 10 – Order matters

y = (x+2)^2 and y = x^2 + 2.

Let $f(x)=x+5$ and $g(x)=3x$. Show that $f(g(x))\neq g(f(x))$.

Exercises 11 and up

Exercise 11 – Domain of a composite

Number line for the domain.

Let $f(x)=\sqrt{x}$ and $g(x)=x-4$. Find the domain of $f(g(x))$.

Exercise 12 – Two composites with a reciprocal

Let $f(x)=\frac{1}{x}$ and $g(x)=x-3$. Write $f(g(x))$ and $g(f(x))$, and give each domain.

Exercise 13 – Domain with a square root inside

Let $f(x)=\sqrt{x}$ and $g(x)=x^{2}-9$. Find the domain of $f(g(x))$.

Exercise 14 – Decompose a power

Write $h(x)=(5x-1)^{4}$ as $f(g(x))$ with simple functions $f$ and $g$.

Exercise 15 – Decompose a root

Write $h(x)=\sqrt{x^{2}+7}$ as $f(g(x))$.

Exercise 16 – Three functions

Let $f(x)=x+1$, $g(x)=2x$ and $h(x)=x^{2}$. Find $f(g(h(3)))$.

Exercise 17 – Coupon and discount

Coupon first, then 20% off.

Aisha buys a jacket priced $\$80$. A coupon takes $\$15$ off, and a sale takes $20\%$ off. Let $c(p)=p-15$ and $s(p)=0.8p$. Find $s(c(80))$ and $c(s(80))$. Which order is cheaper?

Exercise 18 – Spreading ripple

Time to radius to area.

A ripple spreads so its radius is $r(t)=3t$ feet after $t$ seconds. The area is $A(r)=\pi r^{2}$. Write $A(r(t))$, then find the area at $t=4$ seconds to the nearest tenth.

Exercise 19 – Bakery cost model

Hours to shirts to cost.

A Little League booster club prints $n(h)=8h$ shirts in $h$ hours. The cost is $C(n)=12n+150$ dollars. Write $C(n(h))$, find the cost for $6$ hours, and find the hours for a cost of $\$1,206$.

Exercise 20 – Verify two functions undo each other

Let $f(x)=\frac{x-5}{2}$ and $g(x)=2x+5$. Show that $f(g(x))=x$ and $g(f(x))=x$.

Key facts behind this function operations worksheet

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the function operations worksheet as a PDF with the button below and keep it with your notes.

All Precalculus chapters: Precalculus math.

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