1. Inverse Functions and Logarithms
Section 1.6
Inverse Functions and Logarithms

2. A function maps a given input to exactly one output.
ONE-TO-ONE FUNCTIONS
A function maps a given input to exactly one output.
Mappings f and g are both functions.
However, notice that in f, no two inputs take on the same output. This is not true for g.

3. A function maps a given input to exactly one output.
ONE-TO-ONE FUNCTIONS
A function maps a given input to exactly one output.
In symbols,
But

4. A function maps a given input to exactly one output.
ONE-TO-ONE FUNCTIONS
A function maps a given input to exactly one output.
Functions that share this property with f are called one-to-one functions.

5. ONE-TO-ONE FUNCTIONS

6. Is the function one-to-one?
ONE-TO-ONE FUNCTIONS
Is the function one-to-one?
Since two different numbers cannot have the same cube, is one-to-one.
Also, the graph of passes the horizontal line test, thus confirming that it is a one-to-one function.

7. Is the function one-to-one?
ONE-TO-ONE FUNCTIONS
Is the function one-to-one?
This one is tougher to figure out analytically.
However, looking at the graph, we can see immediately that it does not pass the horizontal line test.
Therefore, is not one-to-one.

8. INVERSE FUNCTIONS
One-to-one functions are important because they are precisely the functions that possess inverse functions. An inverse function reverses the mapping of another function. We denote an inverse function as

9. INVERSE FUNCTIONS
Note that if the original function is not one-to-one, then its inverse would not be uniquely defined, and would therefore not be a function at all.
g is not one-to-one, therefore g does not have an inverse

10. Inverse Functions and Logarithms
The remainder of notes were done in class.
Reference Section 1.6 of the textbook.
Assignment: 1.6 pg70 #3-15, 20-26, 33-39, 47-50