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

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

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

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.

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.

Let (x) = x2 + 1 and g(x) = 3x + 5. Find each of the following. USING OPERATIONS ON FUNCTIONS Example 1 Let (x) = x2 + 1 and g(x) = 3x + 5. Find each of the following. (a) Solution First determine (1) = 2 and g(1) = 8. Then use the definition.

Let (x) = x2 + 1 and g(x) = 3x + 5. Find each of the following. USING OPERATIONS ON FUNCTIONS Example 1 Let (x) = x2 + 1 and g(x) = 3x + 5. Find each of the following. (b) Solution First determine that (– 3) = 10 and g(– 3) = – 4. Then use the definition.

Let (x) = x2 + 1 and g(x) = 3x + 5. Find each of the following. USING OPERATIONS ON FUNCTIONS Example 1 Let (x) = x2 + 1 and g(x) = 3x + 5. Find each of the following. (c) Solution

Let (x) = x2 + 1 and g(x) = 3x + 5. Find each of the following. USING OPERATIONS ON FUNCTIONS Example 1 Let (x) = x2 + 1 and g(x) = 3x + 5. Find each of the following. (d) Solution

Let Find each function. (a) Solution Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let Find each function. (a) Solution

Let Find each function. (b) Solution Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let Find each function. (b) Solution

Let Find each function. (c) Solution Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let Find each function. (c) Solution

Let Find each function. (d) Solution Example 2 USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let Find each function. (d) Solution

(e) Give the domains of the functions in parts (a)-(d). USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let Find each function. (e) Give the domains of the functions in parts (a)-(d). 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). USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let Find each function. (e) Give the domains of the functions in parts (a)-(d). Solution Thus, the domain of g is

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

EVALUATING COMBINATIONS OF FUNCTIONS Example 3 If possible, use the given representations of functions  and g to evaluate

(a) EVALUATING COMBINATIONS OF FUNCTIONS Example 3 For ( – g)(–2), although (–2) = – 3, g(–2) is undefined because –2 is not in the domain of g.

(a) EVALUATING COMBINATIONS OF FUNCTIONS Example 3 The domains of  and g include 1, so The graph of g includes the origin, so Thus, is undefined.

EVALUATING COMBINATIONS OF FUNCTIONS Example 3 If possible, use the given representations of functions  and g to evaluate (b) x (x) g(x) – 2 – 3 undefined 1 3 4 9 2 In the table, g(– 2) is undefined. Thus, ( – g)(– 2) is undefined.

EVALUATING COMBINATIONS OF FUNCTIONS Example 3 If possible, use the given representations of functions  and g to evaluate (b) x (x) g(x) – 2 – 3 undefined 1 3 4 9 2

EVALUATING COMBINATIONS OF FUNCTIONS 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.

Solution We use a three-step process. FINDING THE DIFFERENCE QUOTIENT Example 4 Let (x) = 2x2 – 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.

Remember this term when squaring x + h FINDING THE DIFFERENCE QUOTIENT Example 4 Let (x) = 2x2 – 3x. Find and simplify the expression for the difference quotient, Solution Step 2 Find the entire numerator Substitute Remember this term when squaring x + h Square x + h

FINDING THE DIFFERENCE QUOTIENT Example 4 Let (x) = 2x2 – 3x. Find and simplify the expression for the difference quotient, Solution Step 2 Distributive property Combine like terms.

Step 3 Find the difference quotient by dividing by h. FINDING THE DIFFERENCE QUOTIENT Example 4 Let (x) = 2x2 – 3x. Find and simplify the expression for the difference quotient, Solution Step 3 Find the difference quotient by dividing by h. Substitute. Factor out h. Divide.

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).

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.

Solution First find g(2): EVALUATING COMPOSITE FUNCTIONS Example 5 Let (x) = 2x – 1 and g(x) (a) Solution First find g(2): Now find

Let (x) = 2x – 1 and g(x) (b) Solution EVALUATING COMPOSITE FUNCTIONS Example 5 Let (x) = 2x – 1 and g(x) (b) Solution

Find each of the following. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 6 Given that Find each of the following. (a) Solution 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.

Find each of the following. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 6 Given that Find each of the following. (a) Solution Therefore, the domain of

Find each of the following. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 6 Given that Find each of the following. (b) Solution 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

DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 (a) Solution Multiply the numerator and denominator by x.

DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 (a) 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 Example 7 (a) Solution The solution must be excluded. Multiply by x. Divide by 3.

DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 (a) 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 Example 7 (b) Solution Note that this is meaningless if x = 3

DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 (b) 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

Let (x) = 4x + 1 and g(x) = 2x2 + 5x. SHOWING THAT IS NOT EQUIVALENT TO Example 8 Let (x) = 4x + 1 and g(x) = 2x2 + 5x. Solution Square 4x + 1; distributive property.

Let (x) = 4x + 1 and g(x) = 2x2 + 5x. SHOWING THAT IS NOT EQUIVALENT TO Example 8 Let (x) = 4x + 1 and g(x) = 2x2 + 5x. Solution Distributive property. Combine like terms.

Let (x) = 4x + 1 and g(x) = 2x2 + 5x. SHOWING THAT IS NOT EQUIVALENT TO Example 8 Let (x) = 4x + 1 and g(x) = 2x2 + 5x. Solution Distributive property

Find functions  and g such that FINDING FUNCTIONS THAT FORM A GIVEN COMPOSITE Example 9 Find functions  and g such that Solution Note the repeated quantity x2 – 5. If we choose g(x) = x2 – 5 and (x) = x3 – 4x + 3, then There are other pairs of functions  and g that also work.