Section 4.2 Operations with Functions Section 4.2 Operations with Functions

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.