9.5 Inverse Functions

Notes

The inverse $f^{-1}$ undoes $f$: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• To find it: write $y = f(x)$, swap $x$ and $y$, then solve for $y$.
• If $f(a) = b$, then $f^{-1}(b) = a$. The graphs are reflections over the line $y = x$.
• Example: $f(x) = 3x - 6$ gives $x = 3y - 6$, so $\displaystyle f^{-1}(x) = \frac{x + 6}{3}$.
• Exponentials and logarithms are inverses: the inverse of $2^x + 1$ is $\log_2 (x - 1)$.
• A function needs to be one-to-one to have an inverse. Restricting $x \geq 0$ makes $x^2$ invertible, with inverse $\displaystyle \sqrt{x}$.