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Published byJulius Bates Modified over 6 years ago

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**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

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**4.1 - Inverse Functions (page 242-250)**

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**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

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Example 3 (page 244)

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**Determining Whether Two Functions are Inverses**

Two functions are inverses if the meet the following definition.

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**Determining Whether Two Functions are Inverses - Example**

Determine whether f and g are inverse functions

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**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.

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**Horizontal Line Test (page 245)**

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

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**Graphs of Inverse Functions (page 246)**

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**Graphs of Inverse Functions (page 246)**

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**Graphs of Inverse Functions (page 246)**

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**Graphs of Inverse Functions (page 246)**

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**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).

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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.

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**Restricting the Domain to Make Functions Invertible (page 247)**

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**Differentiability Implies Continuity.**

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

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**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.

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