Adding and Subtracting Functions
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 — 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 —
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).
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