Algebra 2 / Unit 1: Functions and Transformations
1.6 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: $\displaystyle y = 3x - 6 \Rightarrow x = 3y - 6 \Rightarrow y = \frac{x + 6}{3}$.
- The graph of $f^{-1}$ is the reflection of $f$ over the line $y = x$.
- The inverse is a function only if $f$ is one-to-one (passes the horizontal line test).
- Check: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Practice
Find the inverse of $f(x) = 3x - 6$.
Swap: $x = 3y - 6$, so $\displaystyle y = \frac{x + 6}{3}$. $f^{-1}$ does not mean reciprocal.
If $f(2) = 9$, then $f^{-1}(9)$ equals:
The inverse swaps inputs and outputs.
Which function's inverse is NOT a function when the domain is all real numbers?
$y = x^2$ fails the horizontal line test: $2^2 = (-2)^2 = 4$.
Ready for more? Take the Unit 1 test.