1. LIAL HORNSBY SCHNEIDER
COLLEGE ALGEBRA
LIAL
HORNSBY
SCHNEIDER

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

3. Operations of Functions
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 /g are defined as follows.
Sum
Difference
Product
Quotient

4. Note The condition g(x) ≠ 0 in the definition of the quotient means that the domain of (/g)(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.

5. Let (x) = x2 + 1 and g(x) = 3x + 5. Find the following.
USING OPERATIONS ON FUNCTIONS
Example 1
Let (x) = x2 + 1 and g(x) = 3x Find the following. a.
Solution Since (1) = 2 and g(1) = 8, use the definition to get

6. Let (x) = x2 + 1 and g(x) = 3x + 5. Find the following.
USING OPERATIONS ON FUNCTIONS
Example 1
Let (x) = x2 + 1 and g(x) = 3x Find the following. b.
Solution Since (– 3) = 10 and g(– 3) = – 4, use the definition to get

7. Let (x) = x2 + 1 and g(x) = 3x + 5. Find the following.
USING OPERATIONS ON FUNCTIONS
Example 1
Let (x) = x2 + 1 and g(x) = 3x Find the following. c.
Solution Since (5) = 26 and g(5) = 20, use the definition to get

8. Let (x) = x2 + 1 and g(x) = 3x + 5. Find the following.
USING OPERATIONS ON FUNCTIONS
Example 1
Let (x) = x2 + 1 and g(x) = 3x Find the following. d.
Solution Since (0) = 1 and g(0) = 5, use the definition to get

9. y
Domains
For functions  and g, the domains of  + g,  – g, and g include all real numbers in the intersections of the domains of  and g, while the domain of /g includes those real numbers in the intersection of the domains of  and g for which g(x) ≠ 0.

10. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS
Example 2
Let a.
Solution

11. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS
Example 2
Let b.
Solution

12. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS
Example 2
Let c.
Solution

13. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS
Example 2
Let d.
Solution

14. e. Give the domains of the functions.
USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS
Example 2
Let
e. Give the domains of the functions.
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 (– , ).

15. e. Give the domains of the functions.
USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS
Example 2
Let
e. Give the domains of the functions.
Solution Since , the domain of g includes just the real numbers that make 2x – 1 nonnegative. Solve 2x – 1  0 to get x  ½ . The domain of g is

16. e. Give the domains of the functions.
USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS
Example 2
Let
e. Give the domains of the functions.
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.
USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS
Example 2
Let
e. Give the domains of the functions.
Solution The domains of includes those real numbers in the intersection 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 3y 9 a. 5
For ( – g)(–2),although (–2) = – 3, g(–2) is undefined because –2 is not in the domain of g. x
– 4
– 2 2 4

20. a. EVALUATING COMBINATIONS OF FUNCTIONS Example 3y 9 a. 5
The domains of  and g include 1, so x
– 4
– 2 2 4

21. a. EVALUATING COMBINATIONS OF FUNCTIONS Example 3y 9 a. 5
The graph of g includes the origin, so x
– 4
– 2 2 4
Thus, 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
In the table, g(– 2) is undefined. Thus, (– g)(– 2) is undefined.

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

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

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

26. EVALUATING COMBINATIONS OF FUNCTIONS
Example 3 c.

27. FINDING THE DIFFERENCE QUOTIENT
Example 4
Let (x) = 2x2 – 3x. Find the difference quotient and simplify the expression.
Solution
Step 1 Find the first term in the numerator, (x + h). Replace the x in (x) with x + h.

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

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

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

31. Caution Notice that (x + h) is not the same as (x) + (h)
Caution Notice that (x + h) is not the same as (x) + (h). For (x) = 2x2 – 3x in Example 4.
but
These expressions differ by 4xh.

32. 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.

33. 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

34. Don’t confuse composition with multiplication
EVALUATING COMPOSITE FUNCTIONS
Example 5
Let (x) = 2x – 1 and g(x) b.
Solution
Don’t confuse composition with multiplication

35. a. Solution 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. a. Solution 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
Divide by 3.
Therefore the domain of is the set of all real numbers except 0 and ⅓, written in interval
notation as

39. b. Solution 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.

41. Solution Therefore, the domain of
DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS
Example 7 b.
Solution Therefore, the domain of
is the set of all real numbers except 3, written

42. Let (x) = 4x + 1 and g(x) = 2x2 + 5x.
SHOWING THAT
Example 8
Let (x) = 4x + 1 and g(x) = 2x2 + 5x.
Solution
Square 4x + 1; distributive property.

43. Let (x) = 4x + 1 and g(x) = 2x2 + 5x.
SHOWING THAT
Example 8
Let (x) = 4x + 1 and g(x) = 2x2 + 5x.
Solution
Distributive property.
Combine terms.

44. Let (x) = 4x + 1 and g(x) = 2x2 + 5x.
SHOWING THAT
Example 8
Let (x) = 4x + 1 and g(x) = 2x2 + 5x.
Solution
Distributive property

45. 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.