Composition of Functions 1

Composition of Functions When two functions are applied in succession, the resulting function is called the composite of the two given functions. One function is inserted within the other in the place of x, and then the resulting function is simplified. The new function is called the explicit function. Symbols: f(g(x)) can also be written as (f o g)(x) g(f(x)) can also be written as (g o f)(x)

Determining a Composition of Two Functions Given f(x) = 8x - 1 and g(x) = 3x2 + 4x, find the following: a. (f o g)(x) This means we sub g(x) into f(x). (f o g)(x) = 8(3x2 + 4x) – 1 =24x2 + 32x – 1 b. (g o f)(x) (g o f)(x) = 3(8x – 1 )2 + 4(8x – 1) =3(8x – 1)(8x -1) + 4(8x – 1) = 3(64x2 – 16x + 1) + 32x – 4 = 192x2 - 48x + 3 + 32x – 4 = 192x2 - 16x - 1

Examples: For each function below, determine possible functions f and g so that y =f(g(x)). a. y = (x + 4)2 b.

Determine domain and range of the composite function y = f o g(x) if f(x) = and g(x) = x – 6.

Evaluating a Composition of Two Functions Given f(x) = 8x - 1 and g(x) = 3x + 4, find the following: b) (g o f)(2) This tells you to find the value of g(x) first, then use this value for the function f(x). Solve for g(2). a) (f o g)(2) (g o f)(x) = g(f(x)) (f o g)(x) = f(g(x)) f(x) = 8x - 1 f(2) = 8(2) - 1 = 15 g(x) = 3x + 4 g(2) = 3(2) + 4 = 10 With the value of g(2) = 10, you now solve for f(10). (g o f)(2) = g(f(2)) = g(15) (f o g)(2) = f(g(2)) = f(10) g(2) = 10 g(x) = 3x + 4 g(15) = 3(15) + 4 = 49 f(x) = 8x - 1 f(10) = 8(10) - 1 = 79 (f o g)(2) = 79 (g o f)(2) = 49 2

Evaluating a Composition of a Function With Itself Given h(x) = 4x + 3, find the following: a) (h o h)(-3) b) (h o h)(x) (h o h)(x) = h(h(x)) = h(4(x) + 3) = h(4x + 3) h(4x + 3) = 4(4x + 3) + 3 = 16x + 12 + 3 = 16x + 15 (h o h)(-3) = h(h(-3)) = h(4(-3) + 3) = h(-9) h(-9) = 4(-9) + 3 = -33 Note: (h o h)(x) ≠ (hh)(x) (hh)(x) = h(x) x h(x) 5.2.3