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Recall that a function is a rule that assigns exactly one output number to each input number.

We can combine functions in several ways in order to make new functions.

Definion â€” Sum and Difference of Two Functions

Given two functions, f(x) and g(x):

The sum of f and g, written (f + g)(x), is defined as (f + g)(x) = f(x) + g(x).

The difference of f and g, written (f - g)(x), is defined as (f - g)(x) = f(x) - g(x).

The domains of (f + g)(x) and (f - g)(x) consist of all real numbers that are in the domain of both f(x) and g(x).

Example 1

Given f(x) = 0.5x + 4 and g(x) = 0.25x + 1, find the sum (f + g)(x).

Solution

 Use the definition for the sum of functions.Substitute for f(x) and g(x). Combine like terms. So, (f + g)(x) = 0.75x + 5. (f + g)(x) = f(x) + g(x) = (0.5x + 4) + (0.25x + 1) = 0.75x + 5

Example 2

Given f(x) = 5x2 + 6x - 12 and g(x) = 8x - 15, find the difference (f - g)(x).

Solution

 Use the definition for the difference of functions. Substitute for f(x) and g(x). Remove parentheses. Combine like terms. (f - g)(x) = f(x) - g(x) = (5x2 + 6x - 12) - (8x - 15) = 5x2 + 6x - 12 - 8x + 15 = 5x2 - 2x + 3
So, (f - g)(x) = 5x2 - 2x + 3.

You have already learned to evaluate a function for a specific value of the input.

 For example, if f(x) = 2x + 1, then we can find f(3) by substituting 3 for x in the function rule. f(x) f(3) = 2x + 1 = 2(3) + 1 = 7
Likewise, we can evaluate the sum or difference of two functions for a given number. There are two methods that are typically used.

Procedure â€” To Evaluate the Sum or Difference of Functions

Step 1 Find (f + g)(x) or (f - g)(x).

Step 2 Use x = a to find (f + g)(a) or (f - g)(a).