solution example 1 add and subtract functions let f (x) = 4x 1/2 and g(x) = –9x 1/2. find the...
TRANSCRIPT
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