Graph f(x) = -1/2x + 1 and its inverse on the same grid.

f^-1(x) = -2x + 2, the mirror image of f across y = x

Graph f(x) = 1/3x + 4 and its inverse on the same grid.

f^-1(x) = 3x - 12, the mirror image of f across y = x

Graph f(x) = -1/2x - 4 and its inverse on the same grid.

f^-1(x) = -2x - 8, the mirror image of f across y = x

Function operations and inverses worksheet for Algebra 2

Combining two functions, and undoing one. Composition is the one worth drilling: f(g(x)) means g happens first, which is the opposite of how it reads.

Also taught in Algebra 1 and Precalculus, each at its own level and with its own examples.

Not quite the one you need? Same words, different skill:

Inverse Functions (Algebra 2). Undoing a function.



Sigma Prep · sigmaprep.io/worksheets Name Function Operations and Inverses Date Period Answer each question. Add, subtract, multiply and divide functions Given $f (x) = 6 x + 1$ and $g (x) = 6 x + 3$ , find $(f + g) (x)$ Given $f (x) = 5 x + 3$ and $g (x) = - 4 x - 7$ , find $(f g) (x)$ Given $f (x) = 4 x + 6$ and $g (x) = 4 x - 8$ , find $(f + g) (x)$ Compose the functions Given $f (x) = - 4 x + 5$ and $g (x) = - 5 x - 3$ , find $f (g (x))$ $f (x) = - 2 x^{2} + 1$ and $g (x) = 2 x - 4$ . Find $g (f (x))$ Given $f (x) = - 4 x + 8$ and $g (x) = - 4 x + 6$ , find $f (g (x))$ $f (x) = x^{2} - 6$ and $g (x) = x + 3$ . Find $f (g (x))$ Given $f (x) = 6 x + 2$ and $g (x) = - 5 x + 5$ , find $f (g (x))$ $f (x) = - 3 x^{2} - 6$ and $g (x) = - x - 6$ . Find $g (f (x))$ Find the inverse Find the inverse of $f (x) = - 14 x - 10$ Find the inverse of $f (x) = {(x + 2)}^{3} - 5$ Find the inverse of $f (x) = 8 x + 2$ Find the inverse of $f (x) = \sqrt[3]{x - 2} - 5$ Find the inverse of $f (x) = - 18 x + 4$ Find the inverse of $f (x) = {(x - 3)}^{3}$ Evaluate a composition from a table Use the table to find $f (g (4))$ . $\begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ f (x) & 3 & 4 & 1 & 3 \\ g (x) & 2 & 4 & 4 & 1 \end{array}$ Use the table to find $g (f (1))$ . $\begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ f (x) & 3 & 2 & 3 & 4 \\ g (x) & 1 & 1 & 1 & 3 \end{array}$ Use the table to find $f (f (2))$ . $\begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ f (x) & 4 & 1 & 4 & 3 \\ g (x) & 2 & 4 & 3 & 2 \end{array}$ Graph a function and its inverse Graph $f (x) = - \frac{1}{2} x + 1$ and its inverse on the same grid. Graph $f (x) = \frac{1}{3} x + 4$ and its inverse on the same grid. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep



Sigma Prep · sigmaprep.io/worksheets Name Function Operations and InversesAnswer keyVersion 1 Date Period Answer each question. Add, subtract, multiply and divide functions Given $f (x) = 6 x + 1$ and $g (x) = 6 x + 3$ , find $(f + g) (x)$ $12 x + 4$ Given $f (x) = 5 x + 3$ and $g (x) = - 4 x - 7$ , find $(f g) (x)$ $- 20 x^{2} - 47 x - 21$ Given $f (x) = 4 x + 6$ and $g (x) = 4 x - 8$ , find $(f + g) (x)$ $8 x - 2$ Compose the functions Given $f (x) = - 4 x + 5$ and $g (x) = - 5 x - 3$ , find $f (g (x))$ $20 x + 17$ $f (x) = - 2 x^{2} + 1$ and $g (x) = 2 x - 4$ . Find $g (f (x))$ $- 4 x^{2} - 2$ Given $f (x) = - 4 x + 8$ and $g (x) = - 4 x + 6$ , find $f (g (x))$ $16 x - 16$ $f (x) = x^{2} - 6$ and $g (x) = x + 3$ . Find $f (g (x))$ $x^{2} + 6 x + 3$ Given $f (x) = 6 x + 2$ and $g (x) = - 5 x + 5$ , find $f (g (x))$ $- 30 x + 32$ $f (x) = - 3 x^{2} - 6$ and $g (x) = - x - 6$ . Find $g (f (x))$ $3 x^{2}$ Find the inverse Find the inverse of $f (x) = - 14 x - 10$ $f^{- 1} (x) = \frac{- x - 10}{14}$ Find the inverse of $f (x) = {(x + 2)}^{3} - 5$ $f^{- 1} (x) = \sqrt[3]{x + 5} - 2$ Find the inverse of $f (x) = 8 x + 2$ $f^{- 1} (x) = \frac{x - 2}{8}$ Find the inverse of $f (x) = \sqrt[3]{x - 2} - 5$ $f^{- 1} (x) = {(x + 5)}^{3} + 2$ Find the inverse of $f (x) = - 18 x + 4$ $f^{- 1} (x) = \frac{- x + 4}{18}$ Find the inverse of $f (x) = {(x - 3)}^{3}$ $f^{- 1} (x) = \sqrt[3]{x} + 3$ Evaluate a composition from a table Use the table to find $f (g (4))$ . $\begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ f (x) & 3 & 4 & 1 & 3 \\ g (x) & 2 & 4 & 4 & 1 \end{array}$ $3$ Use the table to find $g (f (1))$ . $\begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ f (x) & 3 & 2 & 3 & 4 \\ g (x) & 1 & 1 & 1 & 3 \end{array}$ $1$ Use the table to find $f (f (2))$ . $\begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ f (x) & 4 & 1 & 4 & 3 \\ g (x) & 2 & 4 & 3 & 2 \end{array}$ $4$ Graph a function and its inverse Graph $f (x) = - \frac{1}{2} x + 1$ and its inverse on the same grid. $f^{- 1} (x) = - 2 x + 2$ , the mirror image of f across $y = x$ Graph $f (x) = \frac{1}{3} x + 4$ and its inverse on the same grid. $f^{- 1} (x) = 3 x - 12$ , the mirror image of f across $y = x$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Given $f (x) = 2 x + 1$ and $g (x) = x - 4$ , find $f (g (x))$

g happens first, so start with x - 4.
Put that into f in place of x.
2(x - 4) + 1.

The answer is $2 x - 7$ .

Where students go wrong

Doing the functions in the order they are written. f(g(x)) applies g first even though f is on the left. Working the other way gives a different answer nearly every time, which is why the two are worth checking against each other.

Also called combining functions or functions review.

What you can put on this worksheet

Evaluate a function at a value: Given $f (x) = - 3 x + 5$ , find $f (2)$ → $- 1$
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 $f (g (x))$ → $- 3 x$
Find the inverse of a linear function: Find the inverse of $f (x) = 3 x + 4$ → $f^{- 1} (x) = \frac{x - 4}{3}$
At a number: $f (x) = 3 x + 3$ , $g (x) = - 2 x + 1$ . Find $g (f (- 2))$ → $7$
More operations on functions: $f (x) = x^{2} + 3 x + 3$ , $g (x) = - 2 x + 2$ . Find $(f g) (- 1)$ → $4$
Are they inverses?: Are $f (x) = 4 x + 4$ and $g (x) = \frac{x - 4}{4}$ inverses? → Yes
A quadratic outside, or f(f(x)): $f (x) = 2 x^{2} + 3$ and $g (x) = - 2 x + 1$ . Find $f (g (x))$ → $8 x^{2} - 8 x + 5$
From a table: Use the table to find $g (f (2))$ . $\begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ f (x) & 3 & 3 & 3 & 1 \\ g (x) & 2 & 3 & 1 & 4 \end{array}$ → $1$
Cubics, cube roots and square roots: Find the inverse of $f (x) = \sqrt[3]{x - 5} - 5$ → $f^{- 1} (x) = {(x + 5)}^{3} + 5$
Graph a function and its inverse: Graph $f (x) = - \frac{1}{2} x - 4$ and its inverse on the same grid. → $f^{- 1} (x) = - 2 x - 8$ , the mirror image of f across $y = x$
Fractions and decimals in, powers and binomials in, a finite domain: Given $g (x) = - 3 x^{2} + 5 x$ . Find $g (- 5 x)$ → $g (- 5 x) = - 75 x^{2} - 25 x$

Questions about these worksheets

Yes. Every function operations and inverses 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.

Algebra 2 is usually taken around 10th or 11th 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 and inverses 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 12: evaluate a function at a value, add and multiply functions, compose two functions, find the inverse of a linear function, at a number, more operations on functions, are they inverses?, a quadratic outside, or f(f(x)), from a table, cubics, cube roots and square roots, graph a function and its inverse and fractions and decimals in, powers and binomials in, a finite domain. Tick as many as you want and set how many of each, or let it spread them evenly.

Function operations and inverses worksheet for Algebra 2

Select your difficulty

Add, subtract, multiply and divide functions

Compose the functions

Find the inverse

Evaluate a composition from a table

Graph a function and its inverse

Add, subtract, multiply and divide functions

Compose the functions

Find the inverse

Evaluate a composition from a table

Graph a function and its inverse

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these function operations and inverses worksheets free?+

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

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