Inverse Function Finder

Math Algebra • Functions

Written by STEM Calculators Team Published December 13, 2025 Updated February 24, 2026

Enter \(f(x)\) and press Calculate. The tool attempts to solve \(y=f(x)\) for \(x\), producing \(f^{-1}(x)\) when the function is one-to-one on the chosen window.

Blue: \(y=f(x)\). Orange: \(y=f^{-1}(x)\). Gray dashed: \(y=x\). The sample points \((x_0,f(x_0))\) and \((f(x_0),x_0)\) are labeled.

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2. Inverse Function Finder

An inverse function “undoes” the original function. If \(y=f(x)\) is one-to-one (injective), then there exists \(f^{-1}\) such that \[ f\!\left(f^{-1}(x)\right)=x \quad\text{and}\quad f^{-1}\!\left(f(x)\right)=x \] on the appropriate domains.

1) When does an inverse exist?

One-to-one (injective): different inputs give different outputs. If \(x_1\ne x_2\), then \(f(x_1)\ne f(x_2)\).
Horizontal line test: the graph of \(y=f(x)\) is one-to-one on a domain interval if every horizontal line intersects the graph at most once.
Monotonicity guarantee: if \(f\) is strictly increasing or strictly decreasing on an interval, it is one-to-one on that interval.

Many common functions are not one-to-one on all of \(\mathbb{R}\). Example: \(f(x)=x^2\) fails the horizontal line test on \(\mathbb{R}\), but it becomes invertible if you restrict the domain to \(x\ge 0\) (or \(x\le 0\)).

2) The standard method to find \(f^{-1}(x)\)

The textbook method is:

\[ \begin{aligned} \text{(i) } & y=f(x) \\ \text{(ii) } & \text{swap variables: } x=f(y) \\ \text{(iii) } & \text{solve for } y \\ \text{(iv) } & \text{rename } y \to f^{-1}(x) \end{aligned} \]

3) Domain and range swap

The inverse reverses the roles of inputs and outputs:

\[ \mathrm{Dom}\!\left(f^{-1}\right)=\mathrm{Ran}(f),\qquad \mathrm{Ran}\!\left(f^{-1}\right)=\mathrm{Dom}(f). \]

In practice, the range may require analysis (or may depend on domain restrictions).

4) Graph meaning: reflection in \(y=x\)

The graph of \(y=f^{-1}(x)\) is the reflection of \(y=f(x)\) across the line \(y=x\).
A point \((a,b)\) on \(y=f(x)\) becomes \((b,a)\) on \(y=f^{-1}(x)\).
This is why the calculator plots both curves plus the diagonal \(y=x\).

5) Common inverse patterns

\[ \begin{aligned} f(x)=ax+b\ (a\ne 0) &\Rightarrow f^{-1}(x)=\frac{x-b}{a} \\ f(x)=e^x &\Rightarrow f^{-1}(x)=\ln(x)\quad (x>0)\\ f(x)=\ln(x) &\Rightarrow f^{-1}(x)=e^x \\ f(x)=a^x\ (a>0,a\ne 1) &\Rightarrow f^{-1}(x)=\log_a(x)\quad (x>0)\\ f(x)=x^n\ (n\text{ odd}) &\Rightarrow f^{-1}(x)=\sqrt[n]{x} \\ f(x)=x^2 \text{ on } x\ge 0 &\Rightarrow f^{-1}(x)=\sqrt{x} \end{aligned} \]

Inverses involving even powers, absolute values, or non-monotone trigonometric functions typically require domain restriction to be a true function.

6) Verification (why composition matters)

After finding a candidate inverse \(g(x)\), you should verify:

\[ f(g(x))=x \quad\text{and}\quad g(f(x))=x \]

Symbolic check simplifies the compositions as expressions.
Numeric check tests at a sample value \(x_0\), which is useful when domains are subtle.

7) Worked examples

A) Linear example

\[ \begin{aligned} y &= 2x+3 \\ x &= 2y+3 \\ x-3 &= 2y \\ y &= \frac{x-3}{2} \\ f^{-1}(x) &= \frac{x-3}{2} \end{aligned} \]

B) Exponential example (like your screenshot)

\[ \begin{aligned} y &= 2e^{3x-1} \\ x &= 2e^{3y-1} \\ \frac{x}{2} &= e^{3y-1} \\ \ln\!\left(\frac{x}{2}\right) &= 3y-1 \\ y &= \frac{\ln(x/2)+1}{3} \\ f^{-1}(x) &= \frac{\ln(x/2)+1}{3} \end{aligned} \]

8) What the calculator does

Builds the equation \(x=f(y)\) (swap \(x\) and \(y\)) and tries to “peel” operations to solve for \(y\).
Warns if the function does not appear one-to-one on the chosen window (monotonicity check).
Shows \(y=f(x)\), \(y=f^{-1}(x)\), and \(y=x\), and labels the sample point \((x_0,f(x_0))\) and its swapped point \((f(x_0),x_0)\).
Runs a composition verification and a numeric sample check to catch domain issues.

Frequently Asked Questions

What is an inverse function f^{-1}(x)?

An inverse function reverses the input-output mapping of f. If y = f(x), then f^{-1}(y) returns x, which is often written as f(f^{-1}(x)) = x on the appropriate domain.

How do you find an inverse function algebraically?

Write y = f(x), swap x and y, and solve the new equation for y. The resulting expression is f^{-1}(x) after renaming the variable back to x.

Why does the calculator ask for a domain window xmin and xmax?

Many functions are not one-to-one over all real numbers, so an inverse may not be a function unless the domain is restricted. Choosing a window helps enforce a one-to-one interval for inversion and graphing.

How does the horizontal line test relate to having an inverse?

A function is one-to-one on an interval if every horizontal line intersects its graph at most once. If a horizontal line hits the graph more than once on the chosen interval, a single-valued inverse function does not exist there.

What does the line y = x mean on the graph?

The graph of y = f^{-1}(x) is the reflection of y = f(x) across the line y = x. Points swap coordinates, so (a, f(a)) on f corresponds to (f(a), a) on the inverse.

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