Introduction to Inverse Functions

An inverse function is a function that "undoes" the action of another function. If a function $f$ takes an input $x$ and gives you an output $y$, the inverse function (written as $f^{-1}$) takes $y$ and gives you back $x$.

How to Find an Inverse Function

To find the algebraic equation of an inverse function, you can follow a simple process: swap the variables and solve.

Example: Find the inverse of $f(x) = 3x - 7$

1. Replace $f(x)$ with $y$: $y = 3x - 7$
2. Swap $x$ and $y$: $x = 3y - 7$
3. Solve for the new $y$: Add $7$ to both sides: $x + 7 = 3y$ Divide by $3$: $y = \frac{x + 7}{3}$
4. Replace $y$ with $f^{-1}(x)$: $f^{-1}(x) = \frac{x + 7}{3}$

The Horizontal Line Test

Not every function has an inverse. For an inverse to exist, the original function must be one-to-one. This means every output ($y$-value) is paired with exactly one input ($x$-value).

We can test this visually using the Horizontal Line Test: If you can draw a horizontal line anywhere on the graph and it touches the function more than once, the function does not have an inverse.

Example: Does $f(x) = x^2$ have an inverse?

No, it does not. If you look at the graph of $f(x) = x^2$ (a parabola) and draw a horizontal line at $y = 4$, it crosses the graph twice: at $x = 2$ and $x = -2$. Because it fails the horizontal line test, $f(x) = x^2$ does not have an inverse function (unless we restrict its domain to only positive or only negative numbers).

Graphs of Inverse Functions

The graphs of a function and its inverse have a special relationship: they are reflections of each other over the diagonal line $y = x$.

Because we swap the inputs and outputs to find the inverse, if the original function contains the point $(a, b)$, its inverse will always contain the swapped point $(b, a)$.