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 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. Sum Difference Product Quotient

Example 1 Using Operations on Functions

Example 2 Using Operations on Functions Algebraic Solutions (a) (b) (c) (d)

Solution (a) (b) (c) Example 3 Using Operations on Functions

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. (a)Write a linear cost function with x equal to the quantity produced. (b)Find the revenue function if each device sells for $125. (c)Give the profit function for the item. (d)How many items must be sold before the company makes a profit? (e)Support the result with a graphing calculator.

Example 4 Continued Solution (a)Using the slope-intercept form of a line, let (b)Revenue is price  quantity, so (c)Profit = Revenue – Cost (d)Profit must be greater than zero

Given find (a) and (b) Solution (a) (b) Example 5 Composition of Functions

Let and Find (a) and (b) Solution (a) (b) Note: Example 6 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 7 Composition of Functions

Example 8 Composition of Functions The surface area of a sphere S with radius r is S = 4  r 2. (a)Find S(r) that describes the surface area gained when r increases by 2 inches. (b)Determine the amount of extra material needed to manufacture a ball of radius 22 inches as compared to a ball of radius 20 inches.

Example 10

Example 11