1. 3.6 - Inverse Functions
Notation: Say: “f-inverse of x”…
Definition: “For 2 functions f and g, if and then function g is the inverse of function f and vice versa”.
ex: Finding informally:
Verifying algebraically
ex:

2. Verifying algebraically
ex:
Graphs of Inverses
If point (a, b) is on f, then point (b, a) is on f -1 (and vice-versa)
The graph of f -1 is a reflection of the graph of f across the line y = x
ex:

4. f(a) = 3a-2 f(b) = 3b-2 [Set f(a) = f(b)] f(a) = a2 - a 3a-2 = 3b-2
One-to-One Function
A function for which no two elements of the domain of the function correspond to one element of the range.
One-to-one is often written 1-1.
Note: y = f(x) is a function if it passes the vertical line test. It is a 1-1 function if it passes both the vertical line test and the horizontal line test. Another way of testing whether a function is 1-1 is given below:
Test for 1-1 functions:
“If f(a) = f(b) implies that a = b, then f is 1-1.”
Is f(x) = 3x-2 one-to-one?
f(a) = 3a-2
f(b) = 3b-2
[Set f(a) = f(b)]
3a-2 = 3b-2
3a = 3b
a = b implied
Is f(x) = x2 - x one-to-one?
f(a) = a2 - a
f(b) = b2 - b
[Set f(a) = f(b)]
a2 - a = b2 – b
a = b not implied
ex: = (-1)2 - (-1)

5. x3 x2 Horizontal line test
Every horizontal line passes through the graph at only one point… the function is 1-1 x2
A horizontal line passes through the graph at more than one point… the function is not 1-1…

6. Note: A function that is increasing or decreasing over its entire domain is automatically one-to-one…
Finding the Inverse of a Function
Steps:
Replace f(x) with y.
Switch x and y.
If the new equation is not a function, then f does not have an inverse. If the new equation is a function, then solve the new equation for y.
Replace y with f -1.
Verify.

7. ex: