Warm-up for Section 3.2:

0(5). x

7

2(3).

x

x3(4). x

2 3(1). x x 5 3(2). ( )x

23

4(6).

y

x

Warm-up for Section 3.2:

0(5). 1x

75

2(3).

x

xx

334).

1( x

x

2 3 5(1). xx x 5 13 5(2). ( ) xx

23 6

4 8

8

6(6).

x

y

y y

x x

3.1B Homework Answers

1. 28 = 256 2. (-7)3 = -343 3. 1/47 =

1/16384

4. 1/54 = 1/625 5. 1/44 = 1/256 6. 1/86 = 1/262,144

7. 1.342 1012 8. 3.38 10-5 9. 2.025 109

10. 3.73248 10-7 11. 6.6 101 12. 3.5 10-15

13. x4 14. y11 15. 531,441x18

16. 17. 18.

10

1

w 3z

y3

716

n

m

3.1B Homework Answers Continued…

19. 20. 21.

22. 4q3r 23. 24.

25. 26. 27. 1.032

1011

44

3x 632

3x

41449

1

dc 12

9

125g

h2

1

y

5

14

7

4

a

b5

3

9

24

e

f

Operations on Functions

Section 3.2B

Standard: MM2A5d

Essential Question: How do I perform operations with functions?

Vocabulary

Power function: a function of the form y = axb,where a is a real number and b is a rational number

Composition: h(x) = g(f(x)) is the composition of a function g with a function f. • The domain of h is the set of all x-values such that

x is in the domain of f and f(x) is the domain of g.

Investigation 1: New functions can be created from established functions through the operations of addition, subtraction, multiplication, and division. Consider the linear function f(x) = 2x + 1 and the quadratic function g(x) = x2 – 3. Complete the table below for the selected values of the domain. The first column has been done for you.

Table 1: x -2 -1 0 3y = f(x) -3y = g(x) 1

Table 1: x -2 -1 0 3y = f(x) -3y = g(x) 1

Now, keeping the domain fixed, add the range values for f and g to create a new function. Complete the table below to identify the y values for this new function. The first column has been done for you.

Table 2: x -2 -1 0 3y = f(x) + g(x) -2

-1 1 7-2 -3 6

-3 -2 13

This new function is denoted y = f(x) + g(x) or y = (f + g)(x). To find the rule for the new function, simply add the expressions for y = f(x) and y = g(x).

This new function is: y = (2x + 1) + (x2 – 3).In simple form, we have y = x2 + 2x – 2. Let’s call this function h. So, h(x) = x2 + 2x – 2. Evaluate the function for each domain element to check the values in Table 2.  

Let’s call this function h. So, h(x) = x2 + 2x – 2. Evaluate the function for each domain element to check the values in Table 2.  h(-2) = (-2)2 + 2(-2) – 2 = _______h(-1) = (-1)2 + 2(-1) – 2 = _______h(0) = (0)2 + 2(0) – 2 = _______h(3) = (3)2 + 2(3) – 2 = _______

Did you get the same values in Table 2? ________

-2-3-213

YES

Let’s use f(x) = 2x + 1 and g(x) = x2 – 3 and the operations of subtraction, multiplication, and division to create other new functions. Call the new functions s(x), m(x), and d(x), respectively.

(2). f(x) – g(x) or (f – g)(x)

= (2x + 1) – (x2 – 3) = 2x + 1 – x2 + 3 = -x2 + 2x + 4 s(x) = -x2 + 2x + 4

Let’s use f(x) = 2x + 1 and g(x) = x2 – 3 and the operations of subtraction, multiplication, and division to create other new functions. Call the new functions s(x), m(x), and d(x), respectively.

(3). f(x) ∙ g(x) or (f g)(x)

= (2x + 1)(x2 – 3) = 2x3 – 6x + x2 – 3 = 2x3 + x2 – 6x – 3 m(x) = 2x3 + x2 – 6x – 3

Let’s use f(x) = 2x + 1 and g(x) = x2 – 3 and the operations of subtraction, multiplication, and division to create other new functions. Call the new functions s(x), m(x), and d(x), respectively.

(4). or

)(

)(

xf

xg xf

g

12

32

x

x

12

3)(

2

x

xxd

The domain of the new function is the set of values common to original functions. In other words, it is the intersection of the domains of the original functions. The domain of f(x) = 2x + 1 is _____________ and the domain of g(x) = x2 – 3 is ___________.So, the domains for h, s, and m will all be ___________. 

all realsall reals

all reals

 But, the function d was created by division so we must check to see what values of the common domain will make the denominator zero. This value must be excluded. So, the domain of y = d(x) is all reals except __________.x = -½

2x + 1 = 0 2x = -1 x = -1/2

Check for Understanding: Let h(x) = 3x + 1 and p(x) = 2x – 5 Find the following and state the domain. (5). h(x) + p(x) or (h + p)(x) = ____________

Domain: _____________ 

(6). h(x) – p(x) or (h – p)(x) = ___________

Domain: _____________ 

5x – 4

all reals

x + 6

all reals

Check for Understanding: Let h(x) = 3x + 1 and p(x) = 2x – 5 Find the following and state the domain. (7). h(x) ∙ p(x) or (hp)(x) = ____________

Domain: _____________ 

(8). or = ___________

Domain: ______________________ 

6x2 – 13x – 5

all reals

all reals except x = 5/2

)(

)(

xp

xh xp

h

52

13

x

x

Another way of combining two functions is to form the composition of one with the other.  

-234

-145

11625

Df

Dg

Rf

Rg

f(x) = x + 1 g(x) = x2

The composition of g with f can be pictured above.

The new function created maps the domain of f to the range of g. 

-234

-145

11625

Df

Dg

Rf

Rg

f(x) = x + 1 g(x) = x2

If we call this new function h, then the rule for h is h(x) = (x + 1)2

The domain of h is the set of all x-values such That x is in the domain of g and g(x) is in the domain of f.

h(x) = f(g(x)) 

(9). Let f(x) = 6x and g(x) = 3x + 5 find each composition and its domain. 

a. f(g(x)) =

Domain: _________ 

b. g(f(x)) =

Domain: _________ 

f(3x + 5)

= 6(3x + 5)

g(6x)

= 3(6x) + 5

h(x) = 18x + 5all reals

h(x) = 18x + 5all reals

(9). Let f(x) = 6x and g(x) = 3x + 5 find each composition and its domain.  

c. f(f(x)) =

Domain: __________ 

d. g(g(x)) =

Domain: _________

g(3x + 5)

= 3(3x + 5) + 5

h(x) = 9x + 20

= 9x + 15 + 5

f(6x)

= 6(6x)

h(x) = 36x all reals

all reals

(10). Let f(x) = 2x and g(x) = x2 – 3 find each composition and its domain. 

a. f(g(x)) =

Domain = __________

b. g(f(x) =

Domian = _________

f(x2 – 3 )

= 2(x2 – 3 )

h(x) = 2x2 – 6

g(2x )

= (2x)2 – 3

h(x) = 4x2 – 3

all reals

all reals

(10). Let f(x) = 2x and g(x) = x2 – 3 find each composition and its domain. 

c. g(g(x)) =

Domian = _________

g(x2 – 3 )

= (x2 – 3)2 – 3

h(x) = x4 – 6x2 + 6

= (x2 – 3)(x2 – 3) – 3

= x4 – 3x2 – 3x2 + 9 – 3

all reals

Check for Understanding: Let p(x) = 3x + 1 and h(x) = x2 – 4, find each new functions and its domain.  (11). (p + h)(x) = _____________________

Domain: ________________  (12). (h – p)(x) = _____________________

Domain: ________________  (13). (ph)(x) = _____________________

Domain: ________________   

x2 + 3x – 3

x2 – 3x – 5

3x3 + x2 – 12x – 4

all reals

all reals

all reals

 (14). = _____________________

Domain: ___________________

 (15). p(h(x)) = _____________________Domain: ________________

  (16). h(p(x)) = _____________________

Domain: ________________

)(xh

p

all reals

all reals

all reals except x = ±2

3x2 – 11

9x2 + 6x – 3

2

3 1

4

x

x