2.6 Combinations of Functions; Composite Functions Lets say you charge $5 for each lemonade, and it costs you $1 to produce each lemonade. How much profit do you make for each? How could we write cost, revenue and profit as a function?

The Sum of Functions Let f and g be two functions. The sum f + g is the function defined by (f + g)(x) = f(x) + g(x) The domain of f + g is the set of all real numbers that are common (intersection) to the domain of f and the domain of g.

Definitions: Sum, Difference, Product, and Quotient of Functions Let f and g be two functions. The sum of f + g, the difference f – g, the product fg, and the quotient f /g are functions whose domains are the set of all real numbers common (intersection) to the domains of f and g, defined as follows: Sum: (f + g)(x) = f (x)+g(x) Difference:(f – g)(x) = f (x) – g(x) Product:(f g)(x) = f (x) g(x) Quotient: provided g(x) does not equal 0

Example Let f(x) = 2x+1 and g(x) = x Find f + g, f - g, fg, and f/g. Which, if any of these, would have domains other than “all reals”? Solution: f+g = (2x+1) + (x 2 -2) = x 2 +2x-1 f-g = (2x+1) - (x 2 -2) = -x 2 +2x+3 fg = (2x+1)(x 2 -2) = 2x 3 +x 2 -4x-2 x

REVIEW: Given f(x) = 2x –4 and g(x) = x+1, find: f(3) f(a) f(a+1) f(x+1) or? In general:

Lets say you date each candidate three times before you make a decision about them. If each of your friends sets you up with 5 people, how many dates will you go on? How can we write each of these as a function? Number of dates, where x represents one possible candidate: D(x) = Number of possible candidates where x represents one friend: C(x) = Number of total possible outings given the number of friends, x, you have: F(x) =

What we find is that the number of outings is a function of the number of friends we have, or F(x) = D(C(x)), we can write this more elegantly as pronounced, “D of C of x.” We can apply this concept to functions described in any way, for EXAMPLE if: We can find (Make observations.)

The Composition of Functions The composition of the function f with g is denoted by f o g and is defined by the equation (f o g)(x) = f (g(x)). The domain of the composite function f o g is the set of all x such that x is in the domain of g and g(x) is in the domain of f.

Text Example Given f (x) = 3x – 4 and g(x) = x 2 + 6, find: a. (f o g)(x) b. (g o f)(x) c. (g o f)(-2) Solution a. We begin with (f o g)(x), the composition of f with g. Because (f o g)(x) means f (g(x)), we must replace each occurrence of x in the equation for f by g(x). f (x) = 3x – 4 This is the given equation for f. (f o g)(x) = f (g(x)) = 3g(x) – 4 = 3(x 2 + 6) – 4 = 3x – 4 = 3x Replace g(x) with x Use the distributive property. Simplify. Thus, (f o g)(x) = 3x

Solution b. Next, we find (g o f )(x), the composition of g with f. Because (g o f )(x) means g(f (x)), we must replace each occurrence of x in the equation for g by f (x). g(x) = x This is the given equation for g. (g o f )(x) = = Rewrite without composite notation. Replace f (x) (innermost) with 3x – 4. Simplify. Thus, (g o f )(x) =. Notice that (f o g)(x) _______ (g o f )(x). Ex Con’t: Given f (x) = 3x – 4 and g(x) = x 2 + 6, find: b. (g o f)(x)

Ex Con’t: Given f (x) = 3x – 4 and g(x) = x 2 + 6, find: c. (g o f)(-2) Solution c.

Examples P 266 # 45 (f/g & domain only) #66 (find fog & domain) # 78 (opt, but key) find f&g so h(x)=(fog)(x) # 82 #90 Do your homework (and then some)!