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)
.

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

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

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*)
.

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

Subtraction of Functions

The