### 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).

To add two functions, add their outputs. For example, if f (x) = x2 + 2 and g(x) = 4x - 1, then (f + g)(1) = f (1) + g(1) = 3 + 3 = 6. (f + g)(x) = f (x) + g(x) = (x2 +2) + (4x - 1) = x2 + 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).