1. Goal: Find and use inverses of linear and nonlinear functions.
3.4 Inverse Functions
Goal:
Find and use inverses of linear
and nonlinear functions.

2. Inverse Functions
A function and its inverse can be described as the "DO" and the "UNDO" functions. 
A function takes a starting value, performs some operation on this value, and creates an output answer. 
The inverse of this function takes the output answer, performs some operation on it, and arrives back at the original function's starting value.
Domain Range

3. If functions f and g are inverse functions, f(g(x)) = g(f(x)) = x.
This "DO" and "UNDO" process can be stated as a composition of functions.
If functions f and g are inverse functions,
f(g(x)) = g(f(x)) = x.
Example: If f(x) = x-1 and g(x) = x +1
then f(g(x)) = x and g(f(x)) = x
Think of them as "undoing" one another and leaving you right where you started.

4. Definition of an Inverse
 The inverse of a function is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function.        
Notation:  If  f  is a given function, then
f -1 denotes the inverse of  f.

5. Finding the Inverse of a Function
Basically, the process of finding an inverse is simply the swapping of the x and y coordinates. 
This newly formed inverse will be a relation, but may not necessarily be a function – perform horizontal line test of the given function for inverse existence!
One-to-one (1-1) functions are the functions that pass the horizontal line test. If a function fails the horizontal line test, restriction of its domain is required.

6. Three Methods to find the Inverse:
Swapping x and y-values
Solving Algebraically:
a. Set the function = y b.  Swap the x and y variables c.  Solve for y
Reflect graph over the line y = x.

7. Swapping x and y-values
Given relation, find the inverse relation. x -2 -1 1 2 y 4 -4

8. Solving Algebraically
Find the equation of the inverse of the relation f(x) = 2x – 4.
Set the function = y
y = 2x – 4
Swap the x and y variables
x = 2y - 4
Solve for y
y = (x + 4)/2

9. Reflect the graph over y = x
Graph original function
f(x) = 2x + 3
It is drawn in blue.
If reflected over the
Identity line, y = x,
the original function becomes the red
dotted graph. 

10. Practice: Verifying Inverse Functions!!!
Verify that f(x) = 2x – 4 and
f-1(x) = ½x + 2 are inverses.

11. 2. Verify that f and g are inverse functions
f(x) = 3x – 1 and g(x) = ⅓x + ⅓