Not every function has an inverse. For a function to have an inverse, it must be one-to-one, meaning it must pass the horizontal line test. The inverse of a function switches the domain and range and graphs as a reflection across the line y=x. To find the inverse algebraically, replace f(x) with y, switch x and y, and solve for y.

1. Inverse functions
Does every function have an inverse?
No. A function must be 1 to 1 , or pass the
horizontal line test in order to have an inverse.
PASS FAIL
Notation: or
1
f )(1
xf

2. Characteristics of an inverse function
• Algebraically
– The domains and ranges are switched
• Graphically
– The graphs are reflections across the line y = x
21
21
yx
xy

3. How to find an inverse function
1. Verify it passes the HORIZONTAL line test
2. Replace f(x) with y.
3. Switch x and y
4. Solve for y
Example:
Show that and
are inverses a) algebraically b ) graphically
15)( xxf 5
1
)(
x
xg