Precalculus / Unit 1: Functions and Their Graphs
1.5 Inverse Functions
Notes
The inverse $f^{-1}$ undoes $f$: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
- Only one-to-one functions (passing the horizontal line test) have inverses.
- 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 mirror images over $y = x$.
- Example: $f(x) = 3x - 5$. Swap: $x = 3y - 5$, so $\displaystyle f^{-1}(x) = \frac{x + 5}{3}$.
Practice
Find the inverse of $f(x) = 3x - 5$.
Swap to get $x = 3y - 5$, add 5, then divide by 3.
Which function has an inverse that is also a function (over all real numbers)?
Only $x^3$ passes the horizontal line test; the others give the same output for $x$ and $-x$.
If $f(3) = 7$, what is $f^{-1}(7)$?
The inverse reverses the pairing: input 7 returns 3.
Ready for more? Take the Unit 1 test.