Combinations of Functions Section 1.4

Sum, Difference, Product, & Quotient of Functions Let f & g be 2 functions with overlapping domains. then, for all x common to both domains, the sum, difference, product, and quotient of f & g are defined as: 1. Sum ( f+g )(x) = f (x) + g (x) 2. Difference ( f - g )(x) = f (x) - g (x) 3. Product ( f • g )(x) = f (x) • g (x) 4. Quotient ( f / g )(x) = f (x) / g (x), g(x) ≠ 0

Example 1 Given f(x) = 2x + 1 & g(x) = x2 + 2x - 1 Find the sum and difference of the functions. ( f + g)(x) = f(x) + g(x) = (2x+1) + (x2 + 2x - 1) = x2 + 4x ( f - g)(x) = f(x) - g(x) = (2x+1) - (x2 + 2x - 1) = -x2 + 2

Example 1 cont... Find the product of the functions. Given f(x) = 2x + 1 & g(x) = x2 + 2x - 1 Find the product of the functions.

Example 2 Given f(x) = x + 1 & g(x) = x2 - 2x - 3 Find the quotient of the functions. The domain of g(x) is all #’s except 3, -1. Even though it is canceled, this is still the domain.

Example 2 Given & Find the quotient of the functions. The domain of f(x) is all #’s except -1. Even though we canceled, this is still the domain.

You Try Given f(x) = 3x - 2 & g(x) = 3x2 + x - 2 Find the sum and difference of the functions. ( f + g)(x) = f(x) + g(x) = (3x - 2) + (3x2 + x - 2) = 3x2 + 4x - 4 ( f - g)(x) = f(x) - g(x) = (3x - 2) - (3x2 + x - 2) = -3x2 + 2x

You Try Given f(x) = 3x - 2 & g(x) = 3x2 + x - 2 Find the product & quotient of the functions. ( f • g)(x) = f(x) • g(x) = (3x - 2)(3x2 + x - 2) = 9x3 -3x2 -8x + 4 (f ÷ g)(x) = f(x)/g(x) = (3x-2)/(3x2 + x - 2) = 1/(x+1) The domain is still x ≠ 2/3, -1

Composition of Functions Definition of The Composition of Two Functions. The composition of the function f with the function g is (f∘g)(x) The domain of (f∘g)(x) is the set of all x in the domain of g such that g(x) is in the domain of f.

Example 3 Given f(x) = x + 2 & g(x) = 4- x2, find the following. a) f(g(x)) = (4- x2) + 2 = 6 - x2 This means go to the f function and replace x with the g function.

Example 3 Given f(x) = x + 2 and g(x) = 4 - x2, find the following. a) (g ∘f)(x) This means go to the g function and replace x with the f function. g(f(x)) = 4 - (x + 2)2 = 4-(x2 + 4x + 4) = -x2 - 4x

Example 4 cont.. Given f(x) = x + 2 and g(x) = 4 - x2, find the following. b) (g ∘f)(-2) This means go to the f function & replace x with -2. Take that answer & put it into g. f(-2) = -2 + 2 = 0 g(0) = 4 - 02 = 4

Example 4 cont.. Given f(x) = x + 2 and g(x) = 4 - x2, find the following. b) (g ∘f)(-2) You could also go to your answer for (g ∘f)(x) & replace x with a -2 g( f(x)) =-x2 - 4x = -(-2)2 - 4(-2) = 4

You Try Given f(x) = x - 2 & g(x) = x2 + 2x, find the following. a) (f∘g)(x) (x2 + 2x) - 2 = x2 + 2x - 2 b) (g ∘f)(x) (x - 2)2 + 2(x - 2) = x2 - 2x c) (f∘g)(2) (2)2 + 2(2) - 2 = 6 d) (g∘f)(3) (3)2 - 2(3) = 3

Finding the Domain of the Composition of Functions Given f(x) = x2 - 9 & g(x) = √(9-x2), find the composition f(g(x)). Then find the domain of f(g(x)). We take the f function & replace x with the g function. f(g(x)) = ((√ 9-x2))2 - 9 = 9 - x2 - 9 = -x2

Example 5 So the composition f(g(x)) = x2, and the domain of this function is all real numbers. However, this is not the domain of the composition. We need to look at the domain of g(x) to know the domain of the composition. g(x) = √(9-x2), so its domain is where 9 - x2 ≥ 0 g(x) = √(9-x2) 9 - x2 ≥ 0 [-3, 3]