1. Section 4.1: Inverses If the functions f and g satisfy two conditions:
g(f(x)) = x for every x in the domain of f
f(g(x)) = x for every x in the domain of g
then f and g are inverse functions. f is an inverse of g and g is an inverse of f

2. Section 4.1: Inverses Determine whether f and g are inverses:

3. Section 4.1: Inverses
Method for determining the inverse of a function:
Solve for x:

4. Section 4.1: Inverses
A function f has an inverse if and only if its graph is cut at most once by any horizontal line:
A function which passes the vertical and horizontal line tests has an inverse and is one-to-one.

5. Section 4.1: Inverses
Graphically:
If f has an inverse, the graphs of the functions are reflections of one another about the line y = x.

6. Section 4.1: Inverses
If the graph of f is always increasing or decreasing, then the function f has an inverse.
How can we show a function is always increasing or decreasing?

7. Section 4.1: Inverses
The domain of the original is the range of the inverse. The range of the original is the domain of the inverse.