1. Composite Functions Inverse Functions
Section 6.1 Section 6.2
Composite Functions
Inverse Functions

2. THE COMPOSITE FUNCTION
Given two function f and g, the composite function, denote by f ◦ g (read “f composed with g”), is defined by
( f ◦ g)(x) = f (g(x))
The domain of f ◦ g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.

3. CONCEPT OF AN INVERSE FUNCTION
Idea: An inverse function takes the output of the “original” function and tells from what input it resulted.
Note that this really says that the roles of x and y are reversed.

4. MATHEMATICAL DEFINITION OF INVERSE FUNCTIONS
In the language of function notation, two functions f and g are inverses of each other if and only if

5. NOTATION FOR THE INVERSE FUNCTION
We use the notation
for the inverse of f(x).
NOTE:
does NOT mean

6. ONE-TO-ONE FUNCTIONS
A function is one-to-one if for each y-value there is only one x‑value that can be paired with it; that is, each output comes from only one input.

7. ONE-TO-ONE FUNCTIONS AND INVERSE FUNCTIONS
Theorem: A function has an inverse if and only if it is one-to-one.

8. TESTING FOR A ONE-TO-ONE FUNCTION
Horizontal Line Test: A function is one-to-one (and has an inverse) if and only if no horizontal line touches its graph more than once.

9. GRAPHING AN INVERSE FUNCTION
Given the graph of a one-to-one function, the graph of its inverse is obtained by switching x- and y-coordinates.
The resulting graph is reflected about the line y = x.

10. FINDING A FORMULA FOR AN INVERSE FUNCTION
To find a formula for the inverse given an equation for a one-to-one function:
Replace f (x) with y.
Interchange x and y.
Solve the resulting equation for y.
Replace y with f -1(x).