The inverse function $f^{-1}(x)$ reverses what $f(x)$ does. If $f(a) = b$, then $f^{-1}(b) = a$.
Finding an inverse:
- Write $y = f(x)$
- Rearrange to make $x$ the subject
- Replace $x$ with $f^{-1}(x)$ and $y$ with $x$
Example: Find $f^{-1}(x)$ for $f(x) = 3x - 5$. $$y = 3x - 5 \implies 3x = y + 5 \implies x = \frac{y+5}{3}$$ $$f^{-1}(x) = \frac{x+5}{3}$$
Verification: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
The graph of $f^{-1}(x)$ is the reflection of $f(x)$ in the line $y = x$.
Domain and range: the domain of $f^{-1}$ is the range of $f$, and vice versa.
Common error: swapping $x$ and $y$ without rearranging first, giving an expression in $y$ rather than $x$.