In this lesson you’ll learn what an inverse function is, how to find the inverse of a function, and how to check that two functions are inverses of each other.
An inverse function “undoes” what the original function does.
If f(x) turns input x into output y, then the inverse f⁻¹(x) turns that output back into the original input.
Key idea:
f ( f − 1 ( x ) ) = x and f − 1 ( f ( x ) ) = x f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x f ( f − 1 ( x )) = x and f − 1 ( f ( x )) = x How to find the inverse:
Replace f(x) with y
Swap x and y
Solve for y
Replace y with f⁻¹(x)
Example: find the inverse of f(x) = 2x + 3.
y = 2 x + 3 x = 2 y + 3 x − 3 = 2 y y = x − 3 2 \begin{aligned}
y &= 2x + 3 \\[1em]
x &= 2y + 3 \\[1em]
x - 3 &= 2y \\[1em]
y &= \frac{x - 3}{2}
\end{aligned} y x x − 3 y = 2 x + 3 = 2 y + 3 = 2 y = 2 x − 3 So f⁻¹(x) = (x − 3) / 2.
To verify: plug the inverse into the original and vice versa. Both should give x.
1. Find the inverse of f(x) = 3x − 6
y = 3 x − 6 x = 3 y − 6 x + 6 = 3 y y = x + 6 3 \begin{aligned}
y &= 3x - 6 \\[1em]
x &= 3y - 6 \\[1em]
x + 6 &= 3y \\[1em]
y &= \frac{x + 6}{3}
\end{aligned} y x x + 6 y = 3 x − 6 = 3 y − 6 = 3 y = 3 x + 6 So f⁻¹(x) = (x + 6) / 3.
2. Verify they are inverses
f(x) and f⁻¹(x) reflected over y = x x y -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 y = x f(x) f⁻¹(x) (3, 3)
f ( f − 1 ( x ) ) = f ( x + 6 3 ) = 3 ( x + 6 3 ) − 6 = x + 6 − 6 = x ✓ f(f^{-1}(x)) = f\!\left(\frac{x + 6}{3}\right) = 3\!\left(\frac{x + 6}{3}\right) - 6 = x + 6 - 6 = x \;\checkmark f ( f − 1 ( x )) = f ( 3 x + 6 ) = 3 ( 3 x + 6 ) − 6 = x + 6 − 6 = x ✓ 3. Find the inverse of g(x) = x² + 4 (domain restricted to x ≥ 0)
y = x 2 + 4 x = y 2 + 4 x − 4 = y 2 y = x − 4 \begin{aligned}
y &= x^2 + 4 \\[1em]
x &= y^2 + 4 \\[1em]
x - 4 &= y^2 \\[1em]
y &= \sqrt{x - 4}
\end{aligned} y x x − 4 y = x 2 + 4 = y 2 + 4 = y 2 = x − 4 We take the positive root because the domain was restricted to x ≥ 0. So g⁻¹(x) = √(x − 4).
Inverse functions are very useful in real life:
Converting temperature: Fahrenheit to Celsius and back again are inverses.
Undoing a discount: if a store applies 20% off, the inverse tells you the original price.
Finding how long it takes to earn a certain amount: if earnings = rate × time, the inverse finds time given earnings.
In finance: finding the interest rate needed to reach a savings goal.
Example: a phone plan costs c(t) = 25 + 0.12t dollars for t texts. The inverse would tell you how many texts you sent if you know the total bill.
Finding an inverse is mostly algebra: swap x and y, then solve for y. The hardest part is remembering to restrict the domain when needed (like with squares). Practice a few simple linear functions first. They’re the easiest. You’re building a very useful skill for later topics like logarithms and trigonometry.
If $f(x) = 4x - 8$, what is $f^{-1}(x)$? A. $4x + 8$ B. $\frac{x + 8}{4}$ C. $\frac{x - 8}{4}$ D. $\frac{x}{4} + 2$
Which statement is true about inverse functions? A. They always have the same graph B. $f(f^{-1}(x)) = f(x)$ C. $f^{-1}(x) = f(x)$ D. $f(f^{-1}(x)) = x$
A function doubles a number and adds 5. What does its inverse do? A. Subtract 5 and divide by 2 B. Subtract 5 and multiply by 2 C. Add 5 and divide by 2 D. Double and subtract 5
Why do we sometimes need to restrict the domain when finding an inverse? A. To avoid negative numbers B. To make it easier to calculate C. To make the inverse also a function D. To make the graph linear
Find the inverse of $f(x) = \frac{x}{3} + 7$. A. $f^{-1}(x) = 3x - 7$ B. $f^{-1}(x) = 3(x - 7)$ C. $f^{-1}(x) = \frac{x - 7}{3}$ D. $f^{-1}(x) = \frac{x}{3} - 7$
Retry Quiz
