Inverse Functions: How to Find, Graph and Verify an Inverse

What an inverse function does

A function takes an input and produces an output. Its inverse, written $f^{-1}$, runs the process backwards: it takes an output of $f$ and returns the input that produced it. If $f(a) = b$, then $f^{-1}(b) = a$. Doing one and then the other brings you back where you started.

A simple everyday example is temperature conversion. The function $F = \frac{9}{5}C + 32$ converts Celsius to Fahrenheit. Its inverse, $C = \frac{5}{9}(F - 32)$, converts back. Feeding 37 degrees Celsius into the first gives 98.6 degrees Fahrenheit, and feeding 98.6 into the second returns 37.

The notation needs care. The $-1$ in $f^{-1}$ is not an exponent. The inverse function $f^{-1}(x)$ is different from the reciprocal $\frac{1}{f(x)}$. For $f(x) = 2x$, the inverse is $f^{-1}(x) = \frac{x}{2}$, but the reciprocal is $\frac{1}{2x}$.

When does an inverse exist?

Reversing a function only works if each output came from exactly one input. Such a function is called one-to-one, or injective. If two different inputs give the same output, there is no way to know which input to return, so no inverse function exists on that domain.

The function $f(x) = x^2$ fails this test on all real numbers, because $f(3) = f(-3) = 9$. Asked to undo 9, an inverse would not know whether to return 3 or $-3$. Graphically, this is the horizontal line test: a function is one-to-one exactly when no horizontal line crosses its graph more than once. The line $y = 9$ crosses the parabola $y = x^2$ twice, so the parabola fails.

Functions that are always increasing or always decreasing pass automatically. Linear functions with non-zero slope, $x^3$, $e^x$ and $\ln x$ are all one-to-one on their natural domains.

How to find an inverse algebraically

1. Replace $f(x)$ with $y$.
2. Swap $x$ and $y$ everywhere.
3. Solve the new equation for $y$.
4. Replace $y$ with $f^{-1}(x)$ and state its domain, which is the range of $f$.

When $y$ appears more than once after swapping, collect all the $y$ terms on one side and factor $y$ out, exactly as in Example 3.

Restricting the domain

A function that is not one-to-one can still have an inverse if you restrict its domain to a piece where it is. This is how square roots and inverse trigonometric functions are defined.

Restricting $f(x) = x^2$ to $x \ge 0$ keeps only the right half of the parabola, which is one-to-one. Its inverse is $f^{-1}(x) = \sqrt{x}$ for $x \ge 0$. Choosing $x \le 0$ instead would give $-\sqrt{x}$.

Inverse trigonometric functions work the same way. Sine is not one-to-one, so $\arcsin x$ is defined as the inverse of sine restricted to $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. That is why $\arcsin\left(\sin\frac{5\pi}{6}\right) = \frac{\pi}{6}$, not $\frac{5\pi}{6}$: the answer must lie in the restricted interval.

Exponentials and logarithms

The most important inverse pair in precalculus is $e^x$ and $\ln x$: $\ln(e^x) = x$ for every real $x$, and $e^{\ln x} = x$ for $x > 0$. More generally, $\log_b x$ is the inverse of $b^x$. The guide on logarithms builds all of their rules from this fact.

Graphs of inverse functions

Because an inverse swaps inputs and outputs, every point $(a, b)$ on the graph of $f$ becomes the point $(b, a)$ on the graph of $f^{-1}$. Geometrically, swapping coordinates is a reflection in the line $y = x$. So the graph of an inverse is the mirror image of the original graph across that diagonal line.

This gives a quick way to sketch an inverse and a useful check: the graphs of $e^x$ and $\ln x$ are reflections of each other, as are $x^3$ and $\sqrt[3]{x}$. Features swap too. A horizontal asymptote of $f$ becomes a vertical asymptote of $f^{-1}$, which is exactly what happened in Example 3. Plotting both functions with the line $y = x$ on the graphing calculator makes the symmetry clear.

Verifying an inverse

To prove two functions are inverses, compose them in both orders and simplify. If both compositions equal $x$ on the relevant domains, the functions are inverses. For Example 3:

A faster informal check is to pick one input, apply $f$, then apply your inverse and see whether you get the input back. It does not prove the formula is right, but it catches most errors. Composition is explained in more detail in the guide to function composition.

Inverses in calculus

If $f$ is differentiable and one-to-one, the slope of the inverse at a point is the reciprocal of the slope of $f$ at the matching point. If $f(a) = b$ and $f'(a) \neq 0$, then:

For $f(x) = x^3 + x$, we have $f(1) = 2$ and $f'(x) = 3x^2 + 1$, so $f'(1) = 4$ and $(f^{-1})'(2) = \frac{1}{4}$. This works even though finding a formula for $f^{-1}$ would be very difficult. It is the reason the derivative of $\ln x$ is $\frac{1}{x}$, since $\ln$ is the inverse of $e^x$.

Common mistakes

• Treating $f^{-1}(x)$ as $\frac{1}{f(x)}$.
• Forgetting to swap $x$ and $y$, which just rearranges the original function.
• Finding a formula for a function that is not one-to-one without restricting its domain.
• Choosing the wrong sign of a square root after restricting the domain.
• Not stating the domain of the inverse, which should equal the range of the original.

The inverse function calculator shows the swap and solve steps and checks the result by composition.

Inverses from tables and in applications

You do not always need a formula. If a table lists a one-to-one function, its inverse is found by swapping the two columns. Suppose a delivery company charges by weight, and its price table shows that 1 kilogram costs 6 dollars, 2 kilograms cost 9 dollars and 3 kilograms cost 12 dollars. The inverse table answers the opposite question: given what a customer paid, how heavy was the parcel? A payment of 9 dollars means 2 kilograms. If two weights had the same price, the inverse question would have no single answer, which is the one-to-one condition appearing in a practical setting.

Many real formulas are used in both directions. A loan calculator computes the monthly payment from the amount borrowed, and its inverse tells you how much you can borrow for a payment you can afford. A recipe scaled by servings can be inverted to find how many servings a given amount of flour makes. In science, a calibration curve turns a known concentration into an instrument reading, and the inverse turns a measured reading back into a concentration. Each time, the inverse is useful precisely because the original relationship is one-to-one over the range of values that matter.

Seeing inverses this way also makes the algebraic steps feel natural. Swapping $x$ and $y$ is just a compact way of saying that the roles of input and output have been exchanged, and solving for $y$ is answering the reverse question.

Frequently asked questions