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**Combinations of Functions**

Composite Functions

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**Sum, Difference, Product, and Quotient of Functions**

Let f and g be two functions with overlapping domains. Then, for all x common to both domains, the sum, difference, product, and quotient of f and g are defined as: Sum Difference Product Quotient

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Example 1 Given and Find the sum and difference of the functions.

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Example 1 Given and Find the sum and difference of the functions.

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Example 1 Given and Find the sum and difference of the functions.

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Example 1 Given and Find the product and quotient of the functions.

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Example 1 Given and Find the product of the functions.

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**Example 3 Given and Find the quotient of the functions.**

The domain of g(x) is all numbers except 3, -1. Even though we canceled, this is still the domain.

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**Example 3 Given and Find the quotient of the functions.**

The domain of f(x) is all numbers except -1. Even though we canceled, this is still the domain.

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You Try Given and Find the sum and difference of the functions.

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You Try Given and Find the sum and difference of the functions.

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You Try Given and Find the product and quotient of the functions.

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**You Try Given and Find the product and quotient of the functions.**

The domain is still

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**Composition of Functions**

Definition of The Composition of Two Functions. The composition of the function f with the function g is The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.

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**Example 4 Given and , find the following. a)**

This means go to the f function and replace x with the g function.

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**Example 4 Given and , find the following. a)**

This means go to the f function and replace x with the g function.

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**Example 4 Given and , find the following. a)**

This means go to the g function and replace x with the f function.

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**Example 4 Given and , find the following. b)**

This means go to the f function and replace x with -2. Take that answer and put it into g.

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**Example 4 Given and , find the following. b)**

You could also go to your answer for and replace x with a -2.

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You Try Given and , find the following. a) b) c) d)

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You Try Given and , find the following. a) b) c) d)

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**Finding the Domain of the Composition of Functions**

Given and , find the composition f(g(x)). Then find the domain of f(g(x)).

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**Finding the Domain of the Composition of Functions**

Given and , find the composition f(g(x)). Then find the domain of f(g(x)). We take the f function and replace x with the g function.

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Example 5 So the composition , and the domain of this function is all real numbers. However, this is not the domain of the composition. We need to look at the domain of g(x) to know the domain of the composition. , so its domain is where

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Example 5 So the composition , and the domain of this function is all real numbers. However, this is not the domain of the composition. We need to look at the domain of g(x) to know the domain of the composition. , so its domain is where We need to do sign analysis. ___-___|___+___|___-___

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Class work Page 238 6, 8, 14, 16, 20, 22, 36, 42

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Homework Page 238 5-19 odd 35,37,43

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