####
Evaluating Functions

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

Note that functions do not always distribute: *f* (2*x*)≠2*f* (*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*) = 4*x* - 1, then (*f* + *g*)(1) = *f* (1) + *g*(1) = 3 + 3 = 6. (*f* + *g*)(*x*) = *f* (*x*) + *g*(*x*) = (*x*^{2} +2) + (4*x* - 1) = *x*^{2} + 4*x* + 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*):

Addition of Functions

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*) = 2*x* - 1, *g*(*x*) = *x* + 4.

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

Addition of Functions

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*).