Find the inverse of a one–one function using correct notation

Functions & Finding Inverses

What is an Inverse Function?

An inverse function “undoes” the action of the original function. If $f$ takes an input $x$ to an output $y$, then $f^{-1}$ takes that $y$ back to $x$.

Why do we need it?

Think of a vending machine. The machine (function) turns your money (input) into a snack (output). The inverse would be a receipt‑printer that tells you how much money you spent for that snack.

Step‑by‑Step Method

1. Write the function as $y = f(x)$.
2. Swap $x$ and $y$ → $x = f(y)$.
3. Solve this new equation for $y$ (this will be $f^{-1}(x)$).
4. Check that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Example 1: Linear Function

Let $f(x) = 2x + 3$.

?? Check: $f(f^{-1}(x)) = 2\left(\dfrac{x-3}{2}\right)+3 = x$.

Example 2: Quadratic (One‑to‑One Portion)

Take $f(x) = x^2$ but restrict $x \ge 0$ so it’s one‑to‑one.

1. Write $y = x^2$.
2. Swap → $x = y^2$.
3. Take square root: $y = \sqrt{x}$ (choose the positive root because $x \ge 0$).
4. Thus $f^{-1}(x) = \sqrt{x}$.

Practice Problems

• 🔁 Find the inverse of $f(x) = 3x - 7$.
• 🔁 Find the inverse of $f(x) = \dfrac{5}{x}$ (domain $x eq 0$).
• 🔁 Find the inverse of $f(x) = \dfrac{x+4}{2}$.

Quick Tips

• Always check the domain and range before swapping.
• For rational functions, remember to cross‑multiply.
• After solving, plug a value back into both $f$ and $f^{-1}$ to verify.

Remember the Symbol

The inverse is written as $f^{-1}(x)$, not $1/f(x)$. The superscript “−1” is a notation, not a reciprocal.

Your Turn! 🔍

Try to find the inverse of $f(x) = \dfrac{2x+5}{3}$ and check it with a quick test.