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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 2 Graphs and Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc.

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2 2.8 Function Operations and Composition Arithmetic Operations on Functions The Difference Quotient Composition of Functions and Domain

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Operations on Functions and Domains Given two functions and g, then for all values of x for which both (x) and g (x) are defined, the functions + g, – g, g, and are defined as follows. Sum Difference Product Quotient

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 Domains For functions and g, the domains of + g, – g, and g include all real numbers in the intersection of the domains of and g, while the domain of includes those real numbers in the intersection of the domains of and g for which g (x) ≠ 0.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 Note The condition g (x) ≠ 0 in the definition of the quotient means that the domain of (x) is restricted to all values of x for which g (x) is not 0. The condition does not mean that g (x) is a function that is never 0.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 Example 1 USING OPERATIONS ON FUNCTIONS Let (x) = x and g (x) = 3x + 5. Find each of the following. Solution First determine (1) = 2 and g (1) = 8. Then use the definition. (a)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 Example 1 USING OPERATIONS ON FUNCTIONS Let (x) = x and g (x) = 3x + 5. Find each of the following. Solution First determine that (– 3) = 10 and g (– 3) = – 4. Then use the definition. (b)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 Example 1 USING OPERATIONS ON FUNCTIONS Let (x) = x and g (x) = 3x + 5. Find each of the following. Solution (c)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 Example 1 USING OPERATIONS ON FUNCTIONS Let (x) = x and g (x) = 3x + 5. Find each of the following. Solution (d)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Let Find each function. Solution (a)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Solution (b) Let Find each function.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Solution (c) Let Find each function.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Solution (d) Let Find each function.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14 Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Solution To find the domains of the functions, we first find the domains of and g. The domain of is the set of all real numbers (– , ). Because g is defined by a square root radical, the radicand must be non-negative (that is, greater than or equal to 0). (e) Give the domains of the functions in parts (a)-(d). Let Find each function.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Solution (e) Give the domains of the functions in parts (a)-(d). Let Find each function. Thus, the domain of g is

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Solution The domains of + g, – g, g are the intersection of the domains of and g, which is (e) Give the domains of the functions in parts (a)-(d). Let Find each function.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Solution The domain of includes those real numbers in the intersection of the domains for which That is, the domain of is (e) Give the domains of the functions in parts (a)-(d). Let Find each function.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18 Example 3 EVALUATING COMBINATIONS OF FUNCTIONS If possible, use the given representations of functions and g to evaluate

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19 Example 3 EVALUATING COMBINATIONS OF FUNCTIONS (a) For ( – g )(–2), although (–2) = – 3, g (–2) is undefined because –2 is not in the domain of g.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20 Example 3 EVALUATING COMBINATIONS OF FUNCTIONS (a) The domains of and g include 1, so The graph of g includes the origin, so Thus, is undefined.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21 Example 3 If possible, use the given representations of functions and g to evaluate x (x)(x) g(x)g(x) – 2– 2– 3– 3undefined (b) In the table, g (– 2) is undefined. Thus, ( – g )(– 2) is undefined. EVALUATING COMBINATIONS OF FUNCTIONS

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22 Example 3 If possible, use the given representations of functions and g to evaluate x (x)(x) g(x)g(x) – 2– 2– 3– 3undefined (b) EVALUATING COMBINATIONS OF FUNCTIONS

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 23 Example 3 If possible, use the given representations of functions and g to evaluate (c) Using we can find ( f + g )(4) and ( fg )(1). Since –2 is not in the domain of g, ( f – g )(–2) is not defined. EVALUATING COMBINATIONS OF FUNCTIONS

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24 Example 4 FINDING THE DIFFERENCE QUOTIENT Let (x) = 2x 2 – 3x. Find and simplify the expression for the difference quotient, Solution We use a three-step process. Step 1 Find the first term in the numerator, (x + h). Replace x in (x) with x + h.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 25 Example 4 FINDING THE DIFFERENCE QUOTIENT Solution Step 2 Find the entire numerator Substitute Remember this term when squaring x + h Square x + h Let (x) = 2x 2 – 3x. Find and simplify the expression for the difference quotient,

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 26 Example 4 FINDING THE DIFFERENCE QUOTIENT Solution Step 2 Distributive property Combine like terms. Let (x) = 2x 2 – 3x. Find and simplify the expression for the difference quotient,

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 27 Example 4 FINDING THE DIFFERENCE QUOTIENT Solution Step 3 Find the difference quotient by dividing by h. Substitute. Factor out h. Divide. Let (x) = 2x 2 – 3x. Find and simplify the expression for the difference quotient,

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 28 Caution In Example 4, notice that the expression (x + h) is not equivalent to (x) + (h). These expressions differ by 4xh. In general, (x + h) is not equivalent to (x) + (h).

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 29 Composition of Functions and Domain If and g are functions, then the composite function, or composition, of g and is defined by The domain of is the set of all numbers x in the domain of such that (x) is in the domain of g.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 30 Example 5 EVALUATING COMPOSITE FUNCTIONS Let (x) = 2x – 1 and g (x) (a) Solution First find g (2): Now find

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 31 Example 5 EVALUATING COMPOSITE FUNCTIONS Let (x) = 2x – 1 and g (x) (b) Solution

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 32 Example 6 DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS (a) Solution Given that Find each of the following. The domain and range of g are both the set of real numbers. The domain of f is the set of all nonnegative real numbers. Thus, g (x), which is defined as 4x + 2, must be greater than or equal to zero.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 33 Example 6 DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS (a) Solution Given that Find each of the following. Therefore, the domain of

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 34 The domain and range of f are both the set of all nonnegative real numbers. The domain of g is the set of all real numbers. Therefore, the domain of Example 6 DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS (b) Solution Given that Find each of the following.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 35 Example 7 Solution (a) Multiply the numerator and denominator by x. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 36 Example 7 Solution The domain of g is all real numbers except 0, which makes g (x) undefined. The domain of is all real numbers except 3. The expression for g (x), therefore, cannot equal 3. We determine the value that makes g (x) = 3 and exclude it from the domain of DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS (a)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 37 Example 7 Solution The solution must be excluded. Multiply by x. Divide by 3. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS (a)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 38 Example 7 Solution Therefore the domain of is the set of all real numbers except 0 and 1/3, written in interval notation as DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS (a)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 39 Example 7 Solution (b) Note that this is meaningless if x = 3 DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 40 Example 7 Solution The domain of is all real numbers except 3, and the domain of g is all real numbers except 0. The expression for (x), which is, is never zero, since the numerator is the nonzero number 6. Therefore, the domain of is the set of all real numbers except 3, written DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS (b)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 41 Example 8 SHOWING THAT IS NOT EQUIVALENT TO Let (x) = 4x + 1 and g (x) = 2x 2 + 5x. Solution Square 4x + 1; distributive property.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 42 Example 8 Solution Distributive property. Combine like terms. SHOWING THAT IS NOT EQUIVALENT TO Let (x) = 4x + 1 and g (x) = 2x 2 + 5x.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 43 Example 8 Solution Distributive property SHOWING THAT IS NOT EQUIVALENT TO Let (x) = 4x + 1 and g (x) = 2x 2 + 5x.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 44 Example 9 FINDING FUNCTIONS THAT FORM A GIVEN COMPOSITE Find functions and g such that Solution Note the repeated quantity x 2 – 5. If we choose g (x) = x 2 – 5 and (x) = x 3 – 4x + 3, then There are other pairs of functions and g that also work.

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