f(x) = x^2 - x - 4, g(x) = -3x - 4. Find (f/g)(x) and its domain

(x^2 - x - 4)/(-3x - 4), x ≠ -4/3

f(x) = x^2 + x + 3, g(x) = 3x + 4. Find (f/g)(x) and its domain

(x^2 + x + 3)/(3x + 4), x ≠ -4/3

f(x) = x^2 - 2x - 2, g(x) = x - 5. Find (f/g)(x) and its domain

(x^2 - 2x - 2)/(x - 5), x ≠ 5

f(x) = x^2 + x + 1, g(x) = x + 4. Find (f/g)(x) and its domain

(x^2 + x + 1)/(x + 4), x ≠ -4

f(x) = x^2 + 4x - 2, g(x) = x + 5. Find (f/g)(x) and its domain

(x^2 + 4x - 2)/(x + 5), x ≠ -5

Worksheets · Precalculus

Function operations worksheet for Precalculus

Precalculus treats functions as objects to combine: sums, products and quotients with their domains, compositions like f(g(x)), and values such as (f + g)(-1). A quotient f/g loses every x that makes g zero, so the domain is part of the answer, not an extra.

Also taught in Algebra 2, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Function Operations Date Period Answer each question. Add, subtract, multiply and divide functions Given $f (x) = - 8 x + 4$ and $g (x) = x^{2} - 2 x + 6$ , find $(f g) (x)$ $f (x) = x^{2} - x - 4$ , $g (x) = - 3 x - 4$ . Find $(\frac{f}{g}) (x)$ and its domain Given $f (x) = - 7 x - 6$ and $g (x) = x^{2} - 5 x + 1$ , find $(f g) (x)$ $f (x) = x^{2} + x + 3$ , $g (x) = 3 x + 4$ . Find $(\frac{f}{g}) (x)$ and its domain Given $f (x) = 7 x + 11$ and $g (x) = - 3 x^{2} - 2 x + 9$ , find $(f + g) (x)$ $f (x) = x^{2} - 2 x - 2$ , $g (x) = x - 5$ . Find $(\frac{f}{g}) (x)$ and its domain Given $f (x) = - 8 x + 5$ and $g (x) = - 3 x^{2} + 2 x + 5$ , find $(f + g) (x)$ $f (x) = x^{2} + x + 1$ , $g (x) = x + 4$ . Find $(\frac{f}{g}) (x)$ and its domain Given $f (x) = 7 x + 6$ and $g (x) = - 2 x^{2} - 3 x + 5$ , find $(f g) (x)$ $f (x) = x^{2} + 2 x + 4$ , $g (x) = - x + 6$ . Find $(f g) (- 1)$ Given $f (x) = - 9 x + 5$ and $g (x) = - x^{2} + x + 5$ , find $(f + g) (x)$ $f (x) = x^{2} + 4 x - 2$ , $g (x) = x + 5$ . Find $(\frac{f}{g}) (x)$ and its domain Given $f (x) = - 8 x - 7$ and $g (x) = - 3 x^{2} + 4 x - 8$ , find $(f + g) (x)$ Compose the functions Given $f (x) = 8 x - 6$ and $g (x) = - 3 x^{2} - 5 x - 7$ , find $f (g (x))$ Given $f (x) = 7 x - 12$ and $g (x) = 2 x^{2} + 5 x + 1$ , find $f (g (x))$ Given $f (x) = 9 x - 10$ and $g (x) = 3 x^{2} + 4 x - 2$ , find $f (g (x))$ Given $f (x) = 7 x + 10$ and $g (x) = 3 x^{2} - x - 6$ , find $f (g (x))$ Given $f (x) = 8 x - 3$ and $g (x) = x^{2} - 2 x + 2$ , find $f (g (x))$ Given $f (x) = - 7 x - 11$ and $g (x) = 3 x^{2} - x + 2$ , find $f (g (x))$ Given $f (x) = - 7 x + 1$ and $g (x) = x^{2} + 3 x + 1$ , find $f (g (x))$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Function Operations

Date Period

Answer each question.

Add, subtract, multiply and divide functions
Given $f (x) = - 8 x + 4$ and $g (x) = x^{2} - 2 x + 6$ , find $(f g) (x)$
$f (x) = x^{2} - x - 4$ , $g (x) = - 3 x - 4$ . Find $(\frac{f}{g}) (x)$ and its domain
Given $f (x) = - 7 x - 6$ and $g (x) = x^{2} - 5 x + 1$ , find $(f g) (x)$
$f (x) = x^{2} + x + 3$ , $g (x) = 3 x + 4$ . Find $(\frac{f}{g}) (x)$ and its domain
Given $f (x) = 7 x + 11$ and $g (x) = - 3 x^{2} - 2 x + 9$ , find $(f + g) (x)$
$f (x) = x^{2} - 2 x - 2$ , $g (x) = x - 5$ . Find $(\frac{f}{g}) (x)$ and its domain
Given $f (x) = - 8 x + 5$ and $g (x) = - 3 x^{2} + 2 x + 5$ , find $(f + g) (x)$
$f (x) = x^{2} + x + 1$ , $g (x) = x + 4$ . Find $(\frac{f}{g}) (x)$ and its domain
Given $f (x) = 7 x + 6$ and $g (x) = - 2 x^{2} - 3 x + 5$ , find $(f g) (x)$
$f (x) = x^{2} + 2 x + 4$ , $g (x) = - x + 6$ . Find $(f g) (- 1)$
Given $f (x) = - 9 x + 5$ and $g (x) = - x^{2} + x + 5$ , find $(f + g) (x)$
$f (x) = x^{2} + 4 x - 2$ , $g (x) = x + 5$ . Find $(\frac{f}{g}) (x)$ and its domain
Given $f (x) = - 8 x - 7$ and $g (x) = - 3 x^{2} + 4 x - 8$ , find $(f + g) (x)$

Compose the functions
Given $f (x) = 8 x - 6$ and $g (x) = - 3 x^{2} - 5 x - 7$ , find $f (g (x))$
Given $f (x) = 7 x - 12$ and $g (x) = 2 x^{2} + 5 x + 1$ , find $f (g (x))$
Given $f (x) = 9 x - 10$ and $g (x) = 3 x^{2} + 4 x - 2$ , find $f (g (x))$
Given $f (x) = 7 x + 10$ and $g (x) = 3 x^{2} - x - 6$ , find $f (g (x))$
Given $f (x) = 8 x - 3$ and $g (x) = x^{2} - 2 x + 2$ , find $f (g (x))$
Given $f (x) = - 7 x - 11$ and $g (x) = 3 x^{2} - x + 2$ , find $f (g (x))$
Given $f (x) = - 7 x + 1$ and $g (x) = x^{2} + 3 x + 1$ , find $f (g (x))$



Sigma Prep · sigmaprep.io/worksheets Name Function OperationsAnswer keyVersion 2 Date Period Answer each question. Add, subtract, multiply and divide functions Given $f (x) = - 8 x + 4$ and $g (x) = x^{2} - 2 x + 6$ , find $(f g) (x)$ $- 8 x^{3} + 20 x^{2} - 56 x + 24$ $f (x) = x^{2} - x - 4$ , $g (x) = - 3 x - 4$ . Find $(\frac{f}{g}) (x)$ and its domain $\frac{x^{2} - x - 4}{- 3 x - 4}$ , $x \neq - \frac{4}{3}$ Given $f (x) = - 7 x - 6$ and $g (x) = x^{2} - 5 x + 1$ , find $(f g) (x)$ $- 7 x^{3} + 29 x^{2} + 23 x - 6$ $f (x) = x^{2} + x + 3$ , $g (x) = 3 x + 4$ . Find $(\frac{f}{g}) (x)$ and its domain $\frac{x^{2} + x + 3}{3 x + 4}$ , $x \neq - \frac{4}{3}$ Given $f (x) = 7 x + 11$ and $g (x) = - 3 x^{2} - 2 x + 9$ , find $(f + g) (x)$ $- 3 x^{2} + 5 x + 20$ $f (x) = x^{2} - 2 x - 2$ , $g (x) = x - 5$ . Find $(\frac{f}{g}) (x)$ and its domain $\frac{x^{2} - 2 x - 2}{x - 5}$ , $x \neq 5$ Given $f (x) = - 8 x + 5$ and $g (x) = - 3 x^{2} + 2 x + 5$ , find $(f + g) (x)$ $- 3 x^{2} - 6 x + 10$ $f (x) = x^{2} + x + 1$ , $g (x) = x + 4$ . Find $(\frac{f}{g}) (x)$ and its domain $\frac{x^{2} + x + 1}{x + 4}$ , $x \neq - 4$ Given $f (x) = 7 x + 6$ and $g (x) = - 2 x^{2} - 3 x + 5$ , find $(f g) (x)$ $- 14 x^{3} - 33 x^{2} + 17 x + 30$ $f (x) = x^{2} + 2 x + 4$ , $g (x) = - x + 6$ . Find $(f g) (- 1)$ $21$ Given $f (x) = - 9 x + 5$ and $g (x) = - x^{2} + x + 5$ , find $(f + g) (x)$ $- x^{2} - 8 x + 10$ $f (x) = x^{2} + 4 x - 2$ , $g (x) = x + 5$ . Find $(\frac{f}{g}) (x)$ and its domain $\frac{x^{2} + 4 x - 2}{x + 5}$ , $x \neq - 5$ Given $f (x) = - 8 x - 7$ and $g (x) = - 3 x^{2} + 4 x - 8$ , find $(f + g) (x)$ $- 3 x^{2} - 4 x - 15$ Compose the functions Given $f (x) = 8 x - 6$ and $g (x) = - 3 x^{2} - 5 x - 7$ , find $f (g (x))$ $- 24 x^{2} - 40 x - 62$ Given $f (x) = 7 x - 12$ and $g (x) = 2 x^{2} + 5 x + 1$ , find $f (g (x))$ $14 x^{2} + 35 x - 5$ Given $f (x) = 9 x - 10$ and $g (x) = 3 x^{2} + 4 x - 2$ , find $f (g (x))$ $27 x^{2} + 36 x - 28$ Given $f (x) = 7 x + 10$ and $g (x) = 3 x^{2} - x - 6$ , find $f (g (x))$ $21 x^{2} - 7 x - 32$ Given $f (x) = 8 x - 3$ and $g (x) = x^{2} - 2 x + 2$ , find $f (g (x))$ $8 x^{2} - 16 x + 13$ Given $f (x) = - 7 x - 11$ and $g (x) = 3 x^{2} - x + 2$ , find $f (g (x))$ $- 21 x^{2} + 7 x - 25$ Given $f (x) = - 7 x + 1$ and $g (x) = x^{2} + 3 x + 1$ , find $f (g (x))$ $- 7 x^{2} - 21 x - 6$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Function OperationsAnswer keyVersion 2

Date Period

Answer each question.

Add, subtract, multiply and divide functions
Given $f (x) = - 8 x + 4$ and $g (x) = x^{2} - 2 x + 6$ , find $(f g) (x)$ $- 8 x^{3} + 20 x^{2} - 56 x + 24$
$f (x) = x^{2} - x - 4$ , $g (x) = - 3 x - 4$ . Find $(\frac{f}{g}) (x)$ and its domain $\frac{x^{2} - x - 4}{- 3 x - 4}$ , $x \neq - \frac{4}{3}$
Given $f (x) = - 7 x - 6$ and $g (x) = x^{2} - 5 x + 1$ , find $(f g) (x)$ $- 7 x^{3} + 29 x^{2} + 23 x - 6$
$f (x) = x^{2} + x + 3$ , $g (x) = 3 x + 4$ . Find $(\frac{f}{g}) (x)$ and its domain $\frac{x^{2} + x + 3}{3 x + 4}$ , $x \neq - \frac{4}{3}$
Given $f (x) = 7 x + 11$ and $g (x) = - 3 x^{2} - 2 x + 9$ , find $(f + g) (x)$ $- 3 x^{2} + 5 x + 20$
$f (x) = x^{2} - 2 x - 2$ , $g (x) = x - 5$ . Find $(\frac{f}{g}) (x)$ and its domain $\frac{x^{2} - 2 x - 2}{x - 5}$ , $x \neq 5$
Given $f (x) = - 8 x + 5$ and $g (x) = - 3 x^{2} + 2 x + 5$ , find $(f + g) (x)$ $- 3 x^{2} - 6 x + 10$
$f (x) = x^{2} + x + 1$ , $g (x) = x + 4$ . Find $(\frac{f}{g}) (x)$ and its domain $\frac{x^{2} + x + 1}{x + 4}$ , $x \neq - 4$
Given $f (x) = 7 x + 6$ and $g (x) = - 2 x^{2} - 3 x + 5$ , find $(f g) (x)$ $- 14 x^{3} - 33 x^{2} + 17 x + 30$
$f (x) = x^{2} + 2 x + 4$ , $g (x) = - x + 6$ . Find $(f g) (- 1)$ $21$
Given $f (x) = - 9 x + 5$ and $g (x) = - x^{2} + x + 5$ , find $(f + g) (x)$ $- x^{2} - 8 x + 10$
$f (x) = x^{2} + 4 x - 2$ , $g (x) = x + 5$ . Find $(\frac{f}{g}) (x)$ and its domain $\frac{x^{2} + 4 x - 2}{x + 5}$ , $x \neq - 5$
Given $f (x) = - 8 x - 7$ and $g (x) = - 3 x^{2} + 4 x - 8$ , find $(f + g) (x)$ $- 3 x^{2} - 4 x - 15$

Compose the functions
Given $f (x) = 8 x - 6$ and $g (x) = - 3 x^{2} - 5 x - 7$ , find $f (g (x))$ $- 24 x^{2} - 40 x - 62$
Given $f (x) = 7 x - 12$ and $g (x) = 2 x^{2} + 5 x + 1$ , find $f (g (x))$ $14 x^{2} + 35 x - 5$
Given $f (x) = 9 x - 10$ and $g (x) = 3 x^{2} + 4 x - 2$ , find $f (g (x))$ $27 x^{2} + 36 x - 28$
Given $f (x) = 7 x + 10$ and $g (x) = 3 x^{2} - x - 6$ , find $f (g (x))$ $21 x^{2} - 7 x - 32$
Given $f (x) = 8 x - 3$ and $g (x) = x^{2} - 2 x + 2$ , find $f (g (x))$ $8 x^{2} - 16 x + 13$
Given $f (x) = - 7 x - 11$ and $g (x) = 3 x^{2} - x + 2$ , find $f (g (x))$ $- 21 x^{2} + 7 x - 25$
Given $f (x) = - 7 x + 1$ and $g (x) = x^{2} + 3 x + 1$ , find $f (g (x))$ $- 7 x^{2} - 21 x - 6$

How to do these

Given $f (x) = 5 x - 2$ and $g (x) = x^{2} + 3 x - 4$ , find $f (g (x))$

g goes in first: replace x in f with x² + 3x - 4.
f(g(x)) = 5(x² + 3x - 4) - 2 = 5x² + 15x - 20 - 2.

The answer is $5 x^{2} + 15 x - 22$ .

Where students go wrong

Multiplying only the first term. In 5(x² + 3x - 4) the 5 reaches every term, giving 5x² + 15x - 20. Writing 5x² + 3x - 4 is the usual slip in these compositions.

Also called adding functions.

What you can put on this worksheet

Add and multiply functions: Given $f (x) = x + 3$ and $g (x) = - 3 x - 3$ , find $(f + g) (x)$ → $- 2 x$
Compose two functions: Given $f (x) = x + 3$ and $g (x) = - 3 x - 3$ , find $f (g (x))$ → $- 3 x$
More operations on functions: $f (x) = x^{2} + 3 x + 3$ , $g (x) = - 2 x + 2$ . Find $(f g) (- 1)$ → $4$
A composition at an expression: Given $g (x) = 2 x + 2$ and $f (x) = x + 4$ . Find $(g \circ f) (4 + t)$ → $2 t + 18$
Write h as f(g(x)): If $h (x) = f (g (x))$ and $h (x) = \sqrt{x + 6}$ , find $f (x)$ and $g (x)$ , neither equal to $x$ → $f (x) = \sqrt{x}$ , $g (x) = x + 6$
(f/g)(a), sometimes undefined: $f (x) = 2 x^{2} + 3 x - 5$ , $g (x) = - 3 x - 3$ .
Find $(\frac{f}{g}) (5)$ → $- \frac{10}{3}$
pf + qg: $f (x) = 3 x + 3$ , $g (x) = - 2 x^{2} + x - 3$ .
Find $2 f (x) + 2 g (x)$ → $- 4 x^{2} + 8 x$
Operations at an expression: $f (x) = 2 x^{2} + 3$ , $g (x) = - 2 x + 1$ .
Find $(f + g) (- t)$ , multiplied out. → $2 t^{2} + 2 t + 4$

Questions about these worksheets

Yes. Every function operations sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the function operations questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 8: add and multiply functions, compose two functions, more operations on functions, a composition at an expression, write h as f(g(x)), (f/g)(a), sometimes undefined, pf + qg and operations at an expression. Tick as many as you want and set how many of each, or let it spread them evenly.

Applying the functions left to right. f(g(x)) does g first even though f is written first, and doing it the other way gives a different answer nearly every time.

Function operations worksheet for Precalculus

Select your difficulty

Add, subtract, multiply and divide functions

Compose the functions

Add, subtract, multiply and divide functions

Compose the functions

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these function operations worksheets free?+

Do the function operations worksheets come with an answer key?+

Are these function operations worksheets printable?+

What grade are these function operations worksheets for?+

Can I get a different version of the same function operations worksheet?+

Is this a free alternative to Kuta Software for function operations?+

Can I choose how hard the function operations questions are?+

What kinds of question are on a function operations worksheet?+

What do students get wrong with function operations?+