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EXAMPLE 1
Add and subtract functions
Let f (x) = 4x1/2 and g(x) = –9x1/2. Find the following.
a.
f(x) + g(x)
SOLUTION
f (x) + g(x) = 4x1/2 + (–9x1/2) = [4 + (–9)]x1/2 = –5x1/2
b.
f(x) – g(x)
SOLUTION
f (x) – g(x) = 4x1/2 – (–9x1/2) = [4 – (–9)]x1/2 = 13x1/2
EXAMPLE 1
c.
Add and subtract functions
the domains of f + g and f – g
SOLUTION
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.
EXAMPLE 2
Multiply and divide functions
Let f (x) = 6x and g(x) = x3/4. Find the following.
a.
f (x) g(x)
SOLUTION
f (x) g(x) = (6x)(x3/4) = 6x(1 + 3/4) = 6x7/4
b.
f (x)
g(x)
SOLUTION
f (x) 6x
(1 – 3/4) = 6x1/4
=
=
6x
3/4
g(x) x
EXAMPLE 2
c.
Multiply and divide functions
the domains of f
g and f
g
SOLUTION
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
f
domain of g is restricted to all positive real
numbers.
EXAMPLE 3
Solve a multi-step problem
Rhinos
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.
EXAMPLE 3
Solve a multi-step problem
SOLUTION
STEP 1
Find and simplify r(m) s(m).
r(m) s(m) = 241m –0.25 [ (6 106)m0.2 ] Write product of r(m)
and s(m).
= 241(6 106)m(–0.25 + 0.2)
= (1446
106)m –0.05
= (1.446 109)m –0.05
Product of powers
property
Simplify.
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.
1.
f (x) + g(x)
SOLUTION
f (x) + g(x) = –2x2/3 + 7x2/3 = (–2 + 7)x2/3 = 5x2/3
2.
f (x) – g(x)
SOLUTION
f (x) – g(x) = –2x2/3 – 7x2/3 = [–2 + ( –7)]x2/3 = –9x2/3
GUIDED PRACTICE
3.
for Examples 1, 2, and 3
the domains of f + g and f – g
SOLUTION
all real numbers; all real numbers
GUIDED PRACTICE
for Examples 1, 2, and 3
Let f (x) = 3x and g(x) = x1/5. Find the following.
4.
f (x)
g(x)
SOLUTION
5.
3x6/5
f (x)
g(x)
SOLUTION
3x4/5
GUIDED PRACTICE
6.
the domains of f
for Examples 1, 2, and 3
g and f
g
SOLUTION
all real numbers; all real numbers except x=0.
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.
SOLUTION about 7.92 108 heartbeats