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Combination on Functions Given two functions f and g , then for all values of x for which both and are defined, the functions are defined as follows.
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Combination on Functions Given two functions f and g, then for all values of x for which both and are defined, the functions are defined as follows. • The domain of is the intersection of the domains of f and g, while the domain of f /g is the intersection of the domains of f and g for which Sum Difference Product Quotient
Example 2 Using Operations on Functions Algebraic Solutions (a) (b) (c) (d)
Example 3 Using Operations on Functions Solution (a) (b) (c)
Example 4 Using Operations on Functions Finding and Analyzing Cost, Revenue, and Profit Suppose that a businessman invests $1500 as his fixed cost in a new venture that produces and sells a device that makes programming a iPhone easier. Each device costs $100 to manufacture. • Write a linear cost function with x equal to the quantity produced. • Find the revenue function if each device sells for $125. • Give the profit function for the item. • How many items must be sold before the company makes a profit? • Support the result with a graphing calculator.
Example 4 Continued Solution • Using the slope-intercept form of a line, let • Revenue is price quantity, so • Profit = Revenue – Cost • Profit must be greater than zero
Example 5 Composition of Functions Given find (a) and (b) Solution (a) (b)
Example 6 Composition of Functions Let and Find (a) and (b) Solution (a) (b) Note:
Example 7 Composition of Functions • Suppose an oil well off the California coast is leaking. • Leak spreads in circular layer over water • Area of the circle is • At any time t, in minutes, the radius increases 5 feet every minute. • Radius of the circular oil slick is • Express the area as a function of time using substitution.
Example 8 Composition of Functions The surface area of a sphere S with radius r is S = 4 r2. • Find S(r)that describes the surface area gained when r increases by 2 inches. • Determine the amount of extra material needed to manufacture a ball of radius 22 inches as compared to a ball of radius 20 inches.
