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

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

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

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.

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.

6. 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 Find each of the following.
(a)
Solution First determine (1) = 2 and g(1) = 8. Then use the definition.

7. 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 Find each of the following.
(b)
Solution First determine that (– 3) = 10 and g(– 3) = – 4. Then use the definition.

8. 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 Find each of the following.
(c)
Solution

9. 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 Find each of the following.
(d)
Solution

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

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

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

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

14. (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).

15. (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

16. (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

17. (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

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

19. (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.

20. (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.

21. 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.

22. 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

23. 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.

24. 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.

25. 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

26. 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.

27. 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.

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).

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.

30. 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

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

32. 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.

33. 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

34. 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

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

36. 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

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

38. 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

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

40. 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

41. 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.

42. 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.

43. 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

44. 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.