Section 1-2: Composition of Functions If you have two functions, you can form new functions by adding, subtracting, multiplying, or dividing the functions. The following are the notation for these operations. (f + g)(x) (f - g)(x) For each new function, the domain consists of those values of x that are common to both domains f and g. When you divide, the domain is further restricted by excluding any x values that make the denominator g(x) equal to 0. Example 1: Given f(x) = 2x and g(x) = 3x - 2, find each function. a. (f + g)(x) b. (f - g)(x) = f(x) + g(x) = f(x) – g(x) = f(x) g(x) = f(x) + g(x) = 2x x - 2 = 2x 2 + 3x + 3 = f(x) – g(x) = 2x – (3x – 2) = 2x – 3x + 2 = 2x 2 – 3x + 7

c. d. We can use these operations to solve an application problem like example 2. Example 2: BUSINESS Madeline pays $40 to rent a booth at the local craft fair so that she can sell silk flower arrangements. It costs her about $25 to make each arrangement. She sells them for $35 each. a. Write the profit function (Hint: Profit = Revenue minus Cost) b. Find the profit on 5, 10 and, 20 arrangements. = f(x) g(x) = (2x 2 + 5)(3x – 2) = 6x 3 - 4x x - 10 P(x) = r(x) – c(x) P(x) = $35x – ($25x + 40) P(5) = $35(5) – (25(5) + 40)= $175 - $165= $10 P(10) = $35(10) – (25(10) + 40)= $350 - $290= $60 P(20) = $35(20) – (25(20) + 40)= $700 - $540= $160

*Composition of functions: R SS T The function formed by composing two functions f and g is called the ________________ of f and g. A composition is denoted by _____________________ and is read as “f composition g” or “f of g”. A function is performed, and then a second function is performed on the answer from the 1 st function. EX: The answers or ranges from function 1, will now be put into function 2 as the Domains composite

Example 3: Find and for f(x) = 3x 2 + 5x - 1 and g(x) = 4x + 5. = f (4x + 5) = 3(4x + 5) 2 + 5(4x + 5) - 1 = 3(16x x + 25) + 20x = 48x x x + 24 = 48x x + 99 = g (3x 2 + 5x – 1) = 4(3x 2 + 5x – 1) + 5 = 12x x – = 12x x + 1

Example 4: Find and for f(x) = and g(x) =.