Section 4.2
Operations with
Functions
Objectives:
1. To add, subtract, multiply, and
divide functions.
2. To find the composition of
functions.
EXAMPLE 1 Let f(x) = x2 – 9 and g(x)
= x + 3. Find (f + g)(x), (f – g)(x), fg(x), and
f/ (x).
g
(f +g)(x) = f(x) + g(x)
= (x2 – 9) + (x + 3)
= x2 + x – 6
EXAMPLE 1 Let f(x) = x2 – 9 and g(x)
= x + 3. Find (f + g)(x), (f – g)(x), fg(x), and
f/ (x).
g
(f – g)(x) = f(x) – g(x)
= (x2 – 9) – (x + 3)
= x2 – 9 – x – 3
= x2 – x – 12
EXAMPLE 1 Let f(x) = x2 – 9 and g(x)
= x + 3. Find (f + g)(x), (f – g)(x), fg(x), and
f/ (x).
g
(fg)(x) = f(x)g(x)
= (x2 – 9)(x + 3)
= x3 + 3x2 – 9x – 27
EXAMPLE 1 Let f(x) = x2 – 9 and g(x)
= x + 3. Find (f + g)(x), (f – g)(x), fg(x), and
f/ (x).
g
2–9
x
f/ (x) =
g
x+3
(x – 3)(x + 3)
=
x+3
= x – 3, if x ≠ -3
EXAMPLE 2 Let f(x) = 5x – 7 and
g(x) = x2 + 3x – 2. Find f(a + b), f(x2 – 9),
g(4a), and g(3x + 1)
f(a + b) = 5(a + b) – 7
= 5a + 5b – 7
f(x2 – 9) = 5(x2 – 9) – 7
= 5x2 – 45 – 7
= 5x2 – 52
EXAMPLE 2 Let f(x) = 5x – 7 and
g(x) = x2 + 3x – 2. Find f(a + b), f(x2 – 9),
g(4a), and g(3x + 1)
g(4a) = (4a)2 + 3(4a) – 2
= 16a2 + 12a – 2
g(3x + 1) = (3x + 1)2 + 3(3x + 1) – 2
= 9x2 + 6x + 1 + 9x + 3 – 2
= 9x2 + 15x + 2
Definition
Composition An operation that
substitutes the second function into
the first function. In symbols: g ◦ f =
g(f(x)). Read g ◦ f as “the composition
of g with f” or “g composed with f”.
Mapping diagrams provide a useful
representation of composition. Let f(x) = 3x – 5
and g(x) = x2 – 9, and let Df = {5, 3, -1, 0}.
g◦f
-1
0
3
5
Df
f
3x – 5
-8
-5
4
10
Rf
Dg
g
x2 – 9
55
16
7
91
Rg
From the circle diagram you can see that
g ◦ f = {(-1, 55), (0, 16), (3, 7), (5, 91)}.
A function rule for the composition of two
functions could also be used to find the
ordered pairs. The rule can be found from
the rules of the original functions. To find
the rule for the composite function
substitute the second function into the
first as illustrated in Example 3.
Use the rule to check that it obtains the
same set of ordered pairs: {(-1, 55), (0, 16),
(3, 7), (5, 91)}. Check for the ordered pair
(3, 7).
(g ◦ f)(x) = 9x2 – 30x + 16
(g ◦ f)(3) = 9(32) – 30(3) + 16
= 81 – 90 + 16
=7
EXAMPLE 3 Find (g ◦ f)(x) if f(x) =
3x – 5 and g(x) = x2 – 9.
(g ◦ f)(x) = g(f(x))
= g(3x – 5)
= (3x – 5)2 – 9
= 9x2 – 30x + 25 – 9
= 9x2 – 30x + 16
Homework:
pp. 181-182
►A. Exercises
Let f(x) = -2x + 7, g(x) = 5x2, h(x) = x – 9.
Evaluate the following.
3. f(x2)
►A. Exercises
Let f(x) = -2x + 7, g(x) = 5x2, h(x) = x – 9.
Evaluate the following.
5. g(3a + b)
►A. Exercises
If f(x) = -2x + 7, g(x) = 5x2, and h(x) = x – 9,
perform the following operations.
11. fh(x)
►B. Exercises
Let f(x) = x, g(x) = x – 7, h(x) = x2 + 8, k(x) =
5x – 4. Find the function rules for the
composition functions.
19. g ◦ h
►B. Exercises
Let f(x) = x, g(x) = x – 7, h(x) = x2 + 8, k(x) =
5x – 4. Find the function rules for the
composition functions.
23. k ◦ f
■ Cumulative Review
36. Find the amount in a savings account
after five years if $2000 is invested at
5% interest compounded quarterly.
■ Cumulative Review
37. Use the exponential growth function
f(x) = C ● 2x to find the number of
bacteria in a culture after 8 days if
there were originally 20 bacteria.
■ Cumulative Review
38. Graph the piece function
 -1 if x -1
f(x) =  x3 if -1  x  1
 ½x if x  1
■ Cumulative Review
39. Find the slope of a line perpendicular
to 3x + 5y = 6.
■ Cumulative Review
40. Find A for right triangle ABC with
C = 90°, a = 2, and b = 3.