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

4.1 - Inverse Functions(page 242-250)

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 1f x

Example 3(page 244)

3 2

3 2

f x x

y x

3 2x y 2 3 2x y

212

3x y

1 212

3f x x

2Domain :

3

Range : 0

D x x

R f x f x

1

Domain : 0

2Range :

3

D x x

R f x f x

Determining Whether Two Functions are Inverses

Two functions are inverses if the meet the following definition.

Determining Whether Two Functions are Inverses - Example

3 2f x x

212

3g x x

Determine whether f and g are inverse functions

g f x

213 2 2

3x

x2

Domain :3

D x x

f g x

213 2 2

3x

Domain : 0D x x

x

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.

Horizontal Line Test(page 245)

Functions not passing the horizontal line test must have theirdomains restricted in order to work with their inverses.

Graphs of Inverse Functions(page 246)

Graphs of Inverse Functions(page 246)

Graphs of Inverse Functions(page 246)

Graphs of Inverse Functions(page 246)

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).

Example 8(page 247)

for all x.

4However, there is now easy way to solve 5 1, if you switch

and and solve for .

y x

x y y

So, even though we know that f has an inverse, we can notProduce a formula for it.

Restricting the Domain to Make Functions Invertible

(page 247)

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 notnecessarily differentiable. ( Corner point, Point of vertical tangency, or Point of discontinuity.