Inverse functions
Inverse Functions
Inverse Functions is the opening topic of Relations & Functions in the California Common Core State Standards. It is aligned to Standard F-BF.4, which requires students to build inverse functions, verifying them by composition and reading them as reflections across the line y = x.
An inverse function \(f^{-1}\) undoes \(f\), so its graph is the reflection of \(f\) across the line \(y=x\); only one-to-one functions have one.
Theory
The inverse function \(f^{-1}\) reverses \(f\): if \(f(a)=b\), then \(f^{-1}(b)=a\). Consequently:
- \(f\big(f^{-1}(x)\big)=x\) and \(f^{-1}\big(f(x)\big)=x\).
- The graph of \(f^{-1}\) is the reflection of \(f\) across \(y=x\).
- Only a one-to-one function (passes the horizontal line test) has an inverse.
To find an inverse:
How to find an inverse
- Replace \(f(x)\) with \(y\).
- Swap \(x\) and \(y\).
- Solve the new equation for \(y\).
- Write the result as \(f^{-1}(x)\) and, if needed, restrict the domain.
Write \(y=f(x)\), swap \(x\) and \(y\), then solve for \(y\).
| \(y\) | \(=\) | \(2x-3\) |
| \(x\) | \(=\) | \(2y-3\) |
| \(x+3\) | \(=\) | \(2y\) |
| \(f^{-1}(x)\) | \(=\) | \(\dfrac{x+3}{2}\) |
Inverses satisfy \(f(g(x))=x\).
| \(f(g(x))\) | \(=\) | \(2\!\left(\dfrac{x+3}{2}\right)-3\) |
| \(=\) | \((x+3)-3=x\) |
Swap and solve for \(y\).
| \(x\) | \(=\) | \(y^3+1\) |
| \(x-1\) | \(=\) | \(y^3\) |
| \(f^{-1}(x)\) | \(=\) | \(\sqrt[3]{x-1}\) |
\(x^2\) fails the horizontal line test (each output has two inputs). Restricting to \(x\ge 0\) makes it one-to-one, giving the inverse \(\sqrt{x}\).
Common pitfalls
Frequently asked questions
How do you find the inverse of a function?
Swap \(x\) and \(y\) in \(y=f(x)\) and solve for \(y\).
What does the graph of an inverse look like?
It is the reflection of the original graph across the line \(y=x\).
Which functions have an inverse?
Only one-to-one functions — those passing the horizontal line test.
Is \(f^{-1}\) the same as \(\dfrac1f\)?
No. \(f^{-1}\) is the inverse function; \(\dfrac1f\) is the reciprocal.