Section 4.2
Operations with Functions
Section 4.2
Operations with Functions
Objectives:1. To add, subtract, multiply, and
divide functions.2. To find the composition of
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/g(x).
EXAMPLE 1 Let f(x) = x2 – 9 and g(x) = x + 3. Find (f + g)(x), (f – g)(x), fg(x), and f/g(x).
(f +g)(x) = f(x) + g(x)= (x2 – 9) + (x + 3)= x2 + x – 6
(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/g(x).
EXAMPLE 1 Let f(x) = x2 – 9 and g(x) = x + 3. Find (f + g)(x), (f – g)(x), fg(x), and f/g(x).
(f – g)(x) = f(x) – g(x)= (x2 – 9) – (x + 3)= x2 – 9 – x – 3= x2 – x – 12
(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/g(x).
EXAMPLE 1 Let f(x) = x2 – 9 and g(x) = x + 3. Find (f + g)(x), (f – g)(x), fg(x), and f/g(x).
(fg)(x) = f(x)g(x)= (x2 – 9)(x + 3)= x3 + 3x2 – 9x – 27
(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/g(x).
EXAMPLE 1 Let f(x) = x2 – 9 and g(x) = x + 3. Find (f + g)(x), (f – g)(x), fg(x), and f/g(x).
f/g (x) =f/g (x) =x2 – 9
x + 3x2 – 9
x + 3(x – 3)(x + 3)
x + 3(x – 3)(x + 3)
x + 3==
= x – 3, if 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)
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
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)
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
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
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”.
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”.
DefinitionDefinitionDefinitionDefinition
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}.
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}.
-1035
-1035
-8-54
10
-8-54
10
55167
91
55167
91Df Rf
Dg
Rgf
3x – 5g
x2 – 9
g ◦ f
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.
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).
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
(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.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
(g ◦ f)(x) = g(f(x))= g(3x – 5)= (3x – 5)2 – 9= 9x2 – 30x + 25 – 9= 9x2 – 30x + 16
Homework:
pp. 181-182
Homework:
pp. 181-182
►A. ExercisesLet f(x) = -2x + 7, g(x) = 5x2, h(x) = x – 9. Evaluate the following.
3. f(x2)
►A. ExercisesLet f(x) = -2x + 7, g(x) = 5x2, h(x) = x – 9. Evaluate the following.
3. f(x2)
►A. ExercisesLet f(x) = -2x + 7, g(x) = 5x2, h(x) = x – 9. Evaluate the following.
5. g(3a + b)
►A. ExercisesLet f(x) = -2x + 7, g(x) = 5x2, h(x) = x – 9. Evaluate the following.
5. g(3a + b)
►A. ExercisesIf f(x) = -2x + 7, g(x) = 5x2, and h(x) = x – 9, perform the following operations.11. fh(x)
►A. ExercisesIf f(x) = -2x + 7, g(x) = 5x2, and h(x) = x – 9, perform the following operations.11. fh(x)
►B. ExercisesLet 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. ExercisesLet 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. ExercisesLet 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
►B. ExercisesLet 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 Review36. Find the amount in a savings account
after five years if $2000 is invested at 5% interest compounded quarterly.
■ Cumulative Review36. Find the amount in a savings account
after five years if $2000 is invested at 5% interest compounded quarterly.
■ Cumulative Review37. 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 Review37. 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 Review38. Graph the piece function
f(x) =
■ Cumulative Review38. Graph the piece function
f(x) =-1 if x -1x3 if -1 x 1½x if x 1
-1 if x -1x3 if -1 x 1½x if x 1
■ Cumulative Review39. Find the slope of a line perpendicular
to 3x + 5y = 6.
■ Cumulative Review39. Find the slope of a line perpendicular
to 3x + 5y = 6.
■ Cumulative Review40. Find A for right triangle ABC with
C = 90°, a = 2, and b = 3.
■ Cumulative Review40. Find A for right triangle ABC with
C = 90°, a = 2, and b = 3.
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