Not every function has an inverse. A function must be one-to-one, meaning each input is mapped to a unique output, to have an inverse. The inverse of a function switches the domain and range and reflects the graph across the line y=x. To find the inverse algebraically, switch x and y and solve the original equation for y.

2. 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:
f
1
or
f
1
( x)

3. Characteristics of an inverse function
• Algebraically
– The domains and ranges are switched
y1 x2
x1
y2
• Graphically
– The graphs are reflections across the line y = x

4. How to find an inverse function
1. Switch x and y
2. Solve for y
Example:
x 1
Show that f ( x) 5 x 1 and g ( x)
5
are inverses a) algebraically b ) graphically