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# Composition of Functions

Next, we will evaluate the composition of two functions for a given number using the same two methods that we used previously.

Here is an example using Method 1. That is, we will first find the composition of the functions and then evaluate the composition for a specific value of x.

Example 1

Given f(x) = x2 - 9 and g(x) = x + 5, find (f g) when x = 3. That is, find (f g)(3) = f[g(3)].

Solution

 Step 1 Find (f ○ g)(x). Replace g(x) with x = 5. In f(x), replace x with x = 5. Square the binomial. Combine like terms. So, (f ○ g)(x) = x2 + 10x - 16. (f ○ g)(x) = f[g(x)]= f[x + 5] = (x + 5)2 - 9 = x2 + 5x + 5x + 25 - 9 = x2 + 10x - 16 Step 2 Use x = 3 to find (f ○ g)(3). Substitute 3 for x. Simplify. = (3)2 + 10(3) - 16 = 23

So, (f g)(3) = 23.

Now, letâ€™s use Method 2 to evaluate the composition of two functions. Here, we will first evaluate the innermost function for a specific value of x. Then, we will use that result in the outermost function to find the composition.

Example 2

Given f(x) = x2 - 17 and , find (f g) when x = 5. That is, find (f g)(5) = f[g(5)].

Solution

 Step 1 Use x = 5 to find g(5).Substitute 5 for x in g(x). Simplify. g(5) = 3 Step 2 Use the result of step 1 in the function f(x). Substitute 3 for x in f(x). Simplify. f(x) f(3) = x2 - 17 = 32 - 17 = -8

So, (f g)(5) = -8.