College Algebra Twelfth Edition Chapter 2 Graphs and
College Algebra Twelfth Edition Chapter 2 Graphs and Functions Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 1
Section 2. 8 Function Operations and Composition • Arithmetic Operations on Functions • The Difference Quotient • Composition of Functions and Domain Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 2
Operations on Functions and Domains Given two functions f and g, then for all values of x for which both are defined, the functions and are defined as follows. Sum Difference Product Quotient Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 3
Domains For functions f and g, the domains of f + g, f − g, include all real numbers in the intersection of and domains of f and g, while the domain of includes those real numbers in the intersection of the domains of f and g for which Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 4
Note The condition in the definition of the quotient means that the domain of is restricted to all values of x for which is not 0. The condition does not mean that is a function that is never 0. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 5
Example 1: Using Operations on Functions (1 of 4) Let Find each of the following. (a) Solution First determine Then use the definition. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 6
Example 1: Using Operations on Functions (2 of 4) Let Find each of the following. (b) Solution First determine that Then use the definition. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 7
Example 1: Using Operations on Functions (3 of 4) Let Find each of the following. (c) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 8
Example 1: Using Operations on Functions (4 of 4) Let Find each of the following. (d) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 9
Example 2: Using Operations on Functions and Determining Domains (1 of 8) Let Find each function. (a) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 10
Example 2: Using Operations on Functions and Determining Domains (2 of 8) Let Find each function. (b) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 11
Example 2: Using Operations on Functions and Determining Domains (3 of 8) Let Find each function. (c) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 12
Example 2: Using Operations on Functions and Determining Domains (4 of 8) Let Find each function. (d) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 13
The Difference Quotient • Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 24
Example 4: Finding the Difference Quotient (1 of 4) Find and simplify the Let expression for the difference quotient, Solution We use a three-step process. Step 1 Find the first term in the numerator, with x + h. Replace x in Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 25
Example 4: Finding the Difference Quotient (2 of 4) Find and simplify the Let expression for the difference quotient, Solution Step 2 Find the entire numerator Substitute Square x + h Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 26
Example 4: Finding the Difference Quotient (3 of 4) Find and simplify the Let expression for the difference quotient, Solution Step 2 Distributive property Combine like terms. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 27
Example 4: Finding the Difference Quotient (4 of 4) Find and simplify the Let expression for the difference quotient, Solution Step 3 Find the difference quotient by dividing by h. Substitute. Factor out h. Divide. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 28
Caution In Example 4, notice that the expression is not equivalent to These expressions differ by is not equivalent to In general, Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 29
Composition of Functions and Domain If f and g are functions, then the composite function, or composition, of f and g is defined by The domain of is the set of all numbers x in the domain of f such that is in the domain of f. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 30
Example 5: Evaluating Composite Functions (1 of 2) Let (a) Find Solution First find Now find Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 31
Example 5: Evaluating Composite Functions (2 of 2) Let (b) Find Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 32
Example 6: Determining Composite Functions and Their Domains (1 of 3) Given that Find each of the following. (a) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 33
Example 6: Determining Composite Functions and Their Domains (3 of 3) Given that Find each of the following. (b) Solution • Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 35
Example 9: Finding Functions That Form a Given Composite Find functions and g such that Solution Note the repeated quantity If we choose and then There are other pairs of functions and g that also work. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 45
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