Operations with Functions

Addition and Subtraction of Functions

Evaluating Functions

To evaluate a function f (x) , plug the input in for x . For example, if f (x) = 5x + 12 , then f (4) = 5(4) + 12 = 32 . If g(x) = - , then g(3) = - = - 2 .

Note that functions do not always distribute: f (2x)≠2f (x) , and f (x + 1)≠f (x) + f (1) .

Addition of Functions

To add two functions, add their outputs. For example, if f (x) = x ² + 2 and g(x) = 4x - 1 , then (f + g)(1) = f (1) + g(1) = 3 + 3 = 6 . (f + g)(x) = f (x) + g(x) = (x ² +2) + (4x - 1) = x ² + 4x + 1 . We can see why this in true by looking at the graphs of y = f (x) , y = g(x) , and y = (f + g)(x) :

The y -coordinate of each point on the graph of y = (f + g)(x) is the result of adding the y -coordinate of g(x) to the y -coordinate of f (x) . For example, as noted above, f (1) = 3 , g(1) = 3 , and (f + g)(1) = 3 + 3 = 6 .

Here's another example:

f (x) = 2x - 1 , g(x) = x + 4 .
(f + g)(x) = f (x) + g(x) = (2x - 1) + (x + 4) = 3x + 3 :

The slope of f is 2; in other words, as x increases by 1, f (x) increases by 2. The slope of g is 1: as x increases by 1, g(x) increases by 1. Thus, as x increases by 1, f + g increases by 2 + 1 = 3, and the slope of the sum of two linear functions is the sum of their slopes. The y -intercept of f + g is also a combination of the y - intercepts of f and g : -1 + 4 = 3.

Adding two functions is like plotting one function and taking the graph of that function as the new x -axis. Points of the second function are then plotted with respect to the new axis. For example, (2, 3) becomes "over 2," "up 3 from the new axis," or (3, f + 2) .

Addition of functions is commutative and associative: f + g = g + f and (f + g) + h = f + (g + h) .

Subtraction of Functions

To subtract two functions, subtract their outputs. For example, if f (x) = 2x - 1 and g(x) = x + 4 , then (f - g)(2) = f (2) - g(2) = 3 - 6 = - 3 . (f - g)(x) = f (x) - g(x) = (2x - 1) - (x + 4) = x - 5 . Here is a graph of y = f (x) , y = g(x) , and y = (f + g)(x) :

The y -coordinate of each point on the graph of y = (f - g)(x) is the result of subtracting the y -coordinate of g(x) from the y -coordinate of f (x) . For example, as noted above, f (2) = 3 , g(2) = 6 , and (f - g)(2) = 3 - 6 = - 3 .