SOLUTION

EXAMPLE 1Add and subtract functions

Let f (x) = 4x1/2 and g(x) = –9x1/2. Find the following.

a. f(x) + g(x)

f (x) + g(x) = 4x1/2 + (–9x1/2) = [4 + (–9)]x1/2 = –5x1/2

b. f(x) – g(x)

SOLUTION

f (x) – g(x) = [4 – (–9)]x1/2 = 13x1/2= 4x1/2 – (–9x1/2)

SOLUTION

EXAMPLE 1Add and subtract functions

c. the domains of f + g and f – g

The functions f and g each have the same domain: all nonnegative real numbers. So, the domains of f + g and f – g also consist of all nonnegative real numbers.

SOLUTION

EXAMPLE 2 Multiply and divide functions

Let f (x) = 6x and g(x) = x3/4. Find the following.

a. f (x) g(x)

f (x) g(x)

f (x)g(x) =

6xx3/4 = 6x(1 – 3/4) = 6x1/4

= (6x)(x3/4) = 6x(1 + 3/4) = 6x7/4

b. f (x)g(x)

SOLUTION

SOLUTION

EXAMPLE 2 Multiply and divide functions

c. the domains of f g and fg

The domain of f consists of all real numbers, and the domain of g consists of all nonnegative real numbers. So, the domain of f g consists of all nonnegative real numbers. Because g(0) = 0, the domain of is restricted to all positive real numbers.

fg

Rhinos

EXAMPLE 3 Solve a multi-step problem

For a white rhino, heart rate r (in beats per minute) and life span s (in minutes) are related to body mass m (in kilograms) by these functions:

r(m) = 241m–0.25 s(m) = (6 106)m0.2

• Find r(m) s(m).

• Explain what this product represents.

SOLUTION

EXAMPLE 3 Solve a multi-step problem

STEP 1

Find and simplify r(m) s(m).

= 241(6 106)m(–0.25 + 0.2) Product of powers property

= (1.446 109)m –0.05

Write product of r(m) and s(m).

= 241m –0.25 [ (6 106)m0.2 ]

(1446 106)m –0.05= Simplify.

r(m) s(m)

Use scientific notation.

EXAMPLE 3 Solve a multi-step problem

STEP 2

Interpret r(m) s(m).

Multiplying heart rate by life span gives the total number of heartbeats for a white rhino over its entire lifetime.

GUIDED PRACTICE for Examples 1, 2, and 3

Let f (x) = –2x2/3 and g(x) = 7x2/3. Find the following.

f (x) + g(x)1.

SOLUTION

f (x) + g(x) = –2x2/3 + 7x2/3 = (–2 + 7)x2/3 = 5x2/3

f (x) – g(x)2.

SOLUTION

f (x) – g(x) = –2x2/3 – 7x2/3 = [–2 + ( –7)]x2/3 = –9x2/3

GUIDED PRACTICE for Examples 1, 2, and 3

SOLUTION

3. the domains of f + g and f – g

The domains of f and g have the same domain: all non-negative real numbers. So , the domain of f + g and f – g also consist of all non-negative real numbers.

GUIDED PRACTICE for Examples 1, 2, and 3

Let f (x) = 3x and g(x) = x1/5. Find the following.

SOLUTION

SOLUTION

f (x) g(x)4.

f (x) g(x) = 3x x1/5 = 3(x ) 1 + 1/5 = 3x6/5

f (x)5. g(x)

f (x)g(x) =

3xx1/5 = 3(x ) 1 – 1/5 = 3x4/5

GUIDED PRACTICE for Examples 1, 2, and 3

6. f g

the domains of f g and

The domain of f • g consists of all real numbers.

The domain of consists of all real numbers except x = 0.

SOLUTION

f g

GUIDED PRACTICE for Examples 1, 2, and 3

Rhinos

7. Use the result of Example 3 to find a white rhino’s number of heartbeats over its lifetime if its body mass is 1.7 105 kilograms.

about 7.92 108 heartbeatsANSWER