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 Add and subtract functions c. 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) g(x) = 6x x3/4 = 6x(1 – 3/4) = 6x1/4

EXAMPLE 2 Multiply and divide functions f c. the domains of f g and 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 domain of is restricted to all positive real numbers. f g

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.

Solve a multi-step problem 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) Product of powers property (1446 106)m –0.05 = Simplify. = (1.446 109)m –0.05 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) SOLUTION f (x) + g(x) = –2x2/3 + 7x2/3 = (–2 + 7)x2/3 = 5x2/3 f (x) – g(x) SOLUTION f (x) – g(x) = –2x2/3 – 7x2/3 = [–2 + ( –7)]x2/3 = –9x2/3

GUIDED PRACTICE for Examples 1, 2, and 3 the domains of f + g and f – g SOLUTION 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. f (x) g(x) SOLUTION f (x) g(x) = 3x x1/5 = 3(x ) 1 + 1/5 = 3x6/5 f (x) g(x) SOLUTION f (x) g(x) = 3x x1/5 = 3(x ) 1 – 1/5 = 3x4/5

GUIDED PRACTICE for Examples 1, 2, and 3 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 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 heartbeats ANSWER