Inverse Functions and Reflection

Finding an inverse function

The inverse $f^{-1}(x)$ "undoes" $f(x)$. The process: replace $f(x)$ with $y$, swap $x$ and $y$, solve for $y$.

Step-by-step: Find the inverse of $f(x) = 3x - 2$

Step 1: $y = 3x - 2$.

Step 2 — Swap: $x = 3y - 2$.

Step 3 — Solve for $y$: $x + 2 = 3y$ → $y = (x + 2)/3$.

Result: $f^{-1}(x) = \frac{x + 2}{3}$.

Check: $f(f^{-1}(x)) = 3 \cdot \frac{x+2}{3} - 2 = x + 2 - 2 = x$ ✓

Graphical property

$f$ and $f^{-1}$ are reflections across the line $y = x$. Every point $(a, b)$ on $f$ corresponds to $(b, a)$ on $f^{-1}$.

When does an inverse exist?

A function has an inverse only if it's one-to-one (passes the horizontal line test — each $y$-value corresponds to exactly one $x$-value).

Try it in the visualization

Both $f$ and $f^{-1}$ are graphed with the $y = x$ mirror line. Matching point pairs $(a,b) \leftrightarrow (b,a)$ are connected by dashed lines. For $y = 3x - 2$: $x= 3y - 2$, so $y = (x+2)/3$. The inverse $f^{-1}(x) = (x+2)/3$ is the reflection of $f(x) = 3x - 2$ across $y = x$.