How to Find the Inverse of a Function
The inverse undoes what the function did. If f turns 2 into 5, then f⁻¹ turns 5 back into 2. The recipe has four steps and it works for most functions you will see at school.
f(x) = 3x − 7
- Replace f(x) with y: y = 3x − 7.
- Swap x and y: x = 3y − 7.
- Solve for y: x + 7 = 3y, so y = (x + 7)/3.
- Write the result as f⁻¹(x) = (x + 7)/3.
- Check: f(f⁻¹(x)) = 3·(x + 7)/3 − 7 = x ✓
Answer: f⁻¹(x) = (x + 7)/3
One-to-One Functions and the Horizontal Line Test
Only a one-to-one function has an inverse that is itself a function. The horizontal line test is the quick check: if any horizontal line hits the graph twice, two inputs share one output and the function cannot be reversed. f(x) = x² fails because both 3 and −3 give 9.
The fix is to restrict the domain. Keep only x ≥ 0 and x² passes the test, with inverse √x. The restriction is not cosmetic; without it the inverse is not a function.
Inverse of a Rational Function
Rational functions put y on both sides after the swap, so you have to gather every y on one side and factor it out.
f(x) = (2x + 1)/(x − 3)
- Write y = (2x + 1)/(x − 3) and swap: x = (2y + 1)/(y − 3).
- Clear the fraction: x(y − 3) = 2y + 1, so xy − 3x = 2y + 1.
- Collect the y terms: xy − 2y = 3x + 1.
- Factor out y: y(x − 2) = 3x + 1.
- Divide: y = (3x + 1)/(x − 2).
Answer: f⁻¹(x) = (3x + 1)/(x − 2)
Notice that f has a vertical asymptote at x = 3 and a horizontal one at y = 2, while f⁻¹ has them at x = 2 and y = 3. The asymptotes swap places, just like the axes.
Inverse of a Function With a Restricted Domain
f(x) = x² + 4, x ≥ 0
- Write y = x² + 4 and swap: x = y² + 4.
- Solve for y²: y² = x − 4.
- Take the square root. Because the original x was ≥ 0, y is ≥ 0, so we keep the positive root: y = √(x − 4).
- The range of f is y ≥ 4, so the domain of f⁻¹ is x ≥ 4.
Answer: f⁻¹(x) = √(x − 4), x ≥ 4
Domain and range swap places between a function and its inverse. That one fact catches most wrong answers.
A Real Inverse: Temperature
F = (9/5)C + 32 turns Celsius into Fahrenheit. Swap and solve: C = (5/9)(F − 32). Body temperature 98.6°F gives (5/9)(66.6) = 37°C. The check works both ways: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. If either fails, the inverse is wrong.
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: the “swap x and y” ritual without a reason. It works because an inverse undoes the function, so inputs and outputs trade places.
Underrated: the horizontal line test. A function has an inverse only if no horizontal line hits its graph twice.
Plain wrong: saying y = x² has an inverse. It fails the horizontal line test until you restrict the domain, for example x ≥ 0, which gives f⁻¹(x) = √x.
Famous Inverse Pairs
| Function | Inverse | Where you meet it |
|---|---|---|
| eˣ | ln x | Growth and decay |
| 10ˣ | log x | Scientific notation, pH |
| x³ | ∛x | Volume and side length |
| F = 9C/5 + 32 | C = 5(F − 32)/9 | Temperature |
| sin x on [−π/2, π/2] | arcsin x | Angles from ratios |
- Writing f⁻¹(x) and thinking it means 1/f(x). It does not; f⁻¹ is the inverse function.
- Forgetting to restrict the domain of x² or |x| before inverting.
- Skipping the composition check.
Why Students Get Stuck on Inverse Functions, and What a Good Calculator Should Do
Searching for help with inverse functions usually starts late at night with one problem that will not work out. Swapping x and y is easy, but verifying the result and the domain is not. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to find f⁻¹, verify it and understand the mirror image, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Confusing f⁻¹(x) with 1/f(x)Find the inverse of f(x) = 3x - 2↗
- Forgetting to swap x and yFind the inverse of f(x) = (x + 1)/(x - 2)↗
- Not checking that f(f⁻¹(x)) = xFind the inverse of f(x) = x^3 + 1↗
- Missing that a function must be one-to-oneFind the inverse of f(x) = sqrt(x - 1)↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the swap-and-solve steps shown
- Wants a verification line
- Wants the reflection across y = x
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Find the inverse of f(x) = 3x - 2↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.