Algebra II

Inverse Functions

Lesson

The inverse of a function $f$ is the function that undoes what $f$ does. It’s written $f^{-1}(x)$ — read as “f inverse of x.”

The $-1$ here is notan exponent. It’s notation that means “the inverse function.”

To find the inverse:

Replace $f(x)$ with $y$ .
Swap $x$ and $y$ in the equation.
Solve for $y$ .
Rewrite as $f^{-1}(x) = \dots$ .

Worked example 1

f(x) = 2x + 6

y = 2x + 6

Swap x and y:

x = 2y + 6

Solve for y:

x - 6 = 2y

y = \frac{x - 6}{2}

So:

f^{-1}(x) = \frac{x - 6}{2}

Worked example 2

f(x) = 3x - 4

Swap and solve:

x = 3y - 4 \;\Rightarrow\; y = \frac{x + 4}{3}

f^{-1}(x) = \frac{x + 4}{3}

How to type your answer

Type the right-hand side only (the rule for $f^{-1}(x)$ ). Use / for division and parentheses to group. No spaces. Examples: (x-6)/2, (x+4)/3, x-5, (x+1)/4.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

f(x) = x + 5

Problem 2

f(x) = x - 3

Problem 3

f(x) = 2x

Problem 4

f(x) = x/4

Practice

Standard problems matching the lesson.

Problem 5

f(x) = 2x + 6

Problem 6

f(x) = 3x - 4

Problem 7

f(x) = x + 7

Problem 8

f(x) = x - 9

Problem 9

f(x) = 5x + 2

Problem 10

f(x) = 4x - 1

Problem 11

f(x) = -x + 3

Problem 12

f(x) = 6x

Problem 13

f(x) = \frac{x}{3}

Problem 14

f(x) = 2x - 8

Problem 15

f(x) = \frac{x}{2} + 1

Problem 16

f(x) = 7x - 14

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

f(x) = -2x + 5

Problem 18

f(x) = \frac{x + 4}{3}

Problem 19

f(x) = \frac{x - 1}{5}

Problem 20

f(x) = 8 - x

Problem 21

f(x) = -3x - 6

Problem 22

f(x) = \frac{x}{4} - 2

Practice

Standard problems matching the lesson.

Problem 23

Approximate C-to-F: F(c) = 2c + 30. Find F⁻¹(x).

Problem 24

Taxi fare f(x) = 3x + 5 (x = miles). Find the inverse f⁻¹(x).

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Linear price model p(x) = 5x − 12. Find p⁻¹(x).

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