EXAMPLE 1 Add and subtract functions Let f

EXAMPLE 1 Add and subtract functions Let f (x) = 4 x 1/2 and g(x) = – 9 x 1/2. Find the following. a. f(x) + g(x) SOLUTION f (x) + g(x) = 4 x 1/2 + (– 9 x 1/2) = [4 + (– 9)]x 1/2 = – 5 x 1/2 b. f(x) – g(x) SOLUTION f (x) – g(x) = 4 x 1/2 – (– 9 x 1/2) = [4 – (– 9)]x 1/2 = 13 x 1/2

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) = 6 x and g(x) = x 3/4. Find the following. a. f (x) g(x) SOLUTION f (x) b. g(x) = (6 x)(x 3/4) = 6 x(1 + 3/4) = 6 x 7/4 f (x) g(x) SOLUTION f (x) 6 x (1 – 3/4) = 6 x 1/4 = = 6 x 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) = 241 m– 0. 25 s(m) = (6 106)m 0. 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) = 241 m – 0. 25 [ (6 106)m 0. 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) = – 2 x 2/3 and g(x) = 7 x 2/3. Find the following. 1. f (x) + g(x) SOLUTION f (x) + g(x) = – 2 x 2/3 + 7 x 2/3 = (– 2 + 7)x 2/3 = 5 x 2/3 2. f (x) – g(x) SOLUTION f (x) – g(x) = – 2 x 2/3 – 7 x 2/3 = [– 2 + ( – 7)]x 2/3 = – 9 x 2/3

GUIDED PRACTICE for Examples 1, 2, and 3 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) = 3 x and g(x) = x 1/5. Find the following. 4. f (x) g(x) SOLUTION 5. 3 x 6/5 f (x) g(x) SOLUTION 3 x 4/5

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