2. Ch 4 - Logarithmic and Exponential Functions - Overview
4.1 - Inverse Functions
4.2 - Logarithmic and Exponential Functions
4.3 - Derivatives of Logarithmic and Exponential Functions
4.4 - Derivatives of Inverse Trigonometric Functions
4.5 - L’Hopital’s Rule; Indeterminate Forms

3. 4.1 - Inverse Functions (page 242-250)

4. Steps For Finding a Functions Inverse
1. Change f(x) to y
2. Switch x and y
3. Solve for y
4. Replace y with

5. Example 3 (page 244)

6. Determining Whether Two Functions are Inverses
Two functions are inverses if the meet the following
definition.

7. Determining Whether Two Functions are Inverses - Example
Determine whether f
and g are inverse functions

8. Horizontal Line Test (page 245)
The Horizontal Line Test is used to determine whether a function would have an inverse over its natural domain.
If a horizontal line is drawn anywhere through the graph of a function and the horizontal line does not intersect the graph in more that one point, then the function passes the horizontal line test.
When a function passes the horizontal line test, the function referred to as one-to-one function. The function is also said to be invertible.

9. Horizontal Line Test (page 245)
Functions not passing the horizontal line test must have their
domains restricted in order to work with their inverses.

10. Graphs of Inverse Functions (page 246)

11. Graphs of Inverse Functions (page 246)

12. Graphs of Inverse Functions (page 246)

13. Graphs of Inverse Functions (page 246)

14. Increasing or Decreasing Functions Have Inverses (page 246)
If the graph of a function f is always increasing or always decreasing over the domain of f, then the function f has an inverse over its entire natural domain.
The derivative of a function (slopes of the tangent lines) determines whether a function is increasing or decreasing over an interval.
So, the following theorem suggest that we can determine whether or not a function has an inverse over its entire domain (passes the horizontal line test).

15. Example 8 (page 247)
for all x.
So, even though we know that f has an inverse, we can not
Produce a formula for it.

16. Restricting the Domain to Make Functions Invertible (page 247)

17. Differentiability Implies Continuity.
Chapter 3 Review Item
Differentiability Implies Continuity.
BUT
Continuity DOES NOT Imply Differentiability

18. Continuity and Differentiability of Inverse Functions (page 248)
If a function is differentiable over an interval, then it is
continuous over that interval.
If a function is continuous over an interval, it is not
necessarily differentiable. ( Corner point, Point of
vertical tangency, or Point of discontinuity.