Integrated Math 2 / Unit 5: Modeling with Functions
5.6 Inverse Functions
Notes
An inverse function $f^{-1}$ undoes $f$. If $f(a) = b$ then $f^{-1}(b) = a$.
- To find it: write $y = f(x)$, swap $x$ and $y$, then solve for $y$.
- Example: $f(x) = 2x + 6$. Swap: $x = 2y + 6$, so $\displaystyle y = \frac{x - 6}{2}$.
- Check: $f(f^{-1}(x)) = x$.
- A function must be one-to-one to have an inverse that is a function. $y = x^2$ needs a restricted domain such as $x \geq 0$.
Practice
Find the inverse of $f(x) = 3x - 4$.
Swap to get $x = 3y - 4$, so $\displaystyle y = \frac{x + 4}{3}$.
If $f(2) = 7$, which must be true?
An inverse swaps inputs and outputs.
Find the inverse of $f(x) = x^3 + 1$.
Swap: $x = y^3 + 1$, so $y^3 = x - 1$ and $\displaystyle y = \sqrt[3]{x - 1}$.
Ready for more? Take the Unit 5 test.