12 Chapter Chapter 2 Exponential and Logarithmic Functions

The Algebra of Functions; Composite Functions Section 12.1 The Algebra of Functions; Composite Functions

Add, Subtract, Multiply, and Divide Functions. Objective 1 Add, Subtract, Multiply, and Divide Functions.

Algebra of Functions Let f and g be functions. New functions from f and g are 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

Example If f(x) = 4x + 3 and g(x) = x2, find a. (f + g)(x) b. (f – g)(x) c. (f · g)(x) d. a. (f + g)(x) = f(x) + g(x) = (4x + 3) + (x2) = x2 + 4x + 3 b. (f – g)(x) = f(x) −g(x) = (4x + 3) – (x2) = −x2 + 4x + 3

Example continued c. (f · g)(x) = f(x) • g(x) = (4x + 3)(x2) where x  0

Construct Composite Functions Objective 2 Construct Composite Functions

We can also combine functions through a function composition. A function composition uses the output from the first function as the input to the second function.

Composition of a Function The composition of function f and g is This means the value of x is first substituted into the function g. Then the value that results from the function g is input into the function f.

Function Composition Notice, that with function composition, we actually activate the functions from right to left in the notation. The function named on the right side of the composition notation is the one we substitute the value for the variable into first.

Example If f(x) = 2x − 5 and g(x) = x2 + 1, then find f(g(2)) = f(5) = 2(5) – 5 = 5 g(f(2)) = g(−1) = (−1)2 + 1 = 2

Example If f(x) = 4x + 3 and g(x) = x2, then find a. b. a. f(g(x)) = 4(x2) + 3 = 4x2 + 3 b. g(f(x)) = (4x + 3)2 = 16x2 + 24x + 9 Notice the results are different with a different order.

Example If f(x) = 7x and g(x) = x – 4, and h(x) = write each function as a composition using two of the given functions. a. b. a. Notice the order in which the function F operates on an input value x. First, 4 is subtracted from x. This is the function g(x) = x – 4. The square root of that result is taken. The square root function is h(x) = . This means Check:

Example continued If f(x) = 7x and g(x) = x – 4, and h(x) = write each function as a composition using two of the given functions. b. Notice the order in which the function G operates on an input value x. First, x is multiplied by 7, and then 4 is subtracted from the result. This means Check: