A * function* is a rule which assigns a unique
output to each input.

* Example.* In mathematics, functions are often
denoted using notation like the following:

This says that the name of the function is f, and x denotes a typical input. The right side tells how to produce an output from the input: Square it. Thus,

Thus, different inputs can produce the same output.

Here is a * table* of some values of f:

Similarly,

Likewise, suppose . Then

Moreover,

The * domain* of a function is the set of
allowable inputs.

* Example.* Suppose .
I can't plug in , as I saw above. Any other number may be
plugged in for x. The domain consists of all x except
--- sometimes I'll get lazy and write this as " " for short.

* Example.* Suppose . Then

Plugging in or would cause division by 0; other values of x are okay. Hence, the domain consists of all real numbers except for 2 and -1.

You can * graph* a function by making a table of
some function values, then plotting the points.

* Example.* Here's the graph of :

Here's the graph of :

Here's the graph of :

is not a function, since some inputs produce more than one output. For example, gives or .

*A graph represents a function if and only if every vertical line
hits the graph at most once.*

Here's the graph of :

This is also not the graph of a function.

Finally, let

The graph of h looks like this:

The * range* of a function is the set of possible outputs (y-values, or
heights).

* Example.* Consider the function .

The graph attains every y-value greater than or equal to 0. Therefore, the range is .

* Example.* Consider the function

The graph attains every y-value greater than or equal to 0 together with . The range is and .

* Example.* Consider the function *whose
entire graph* is shown below.

The function attains every y-value between -1 and 3. It attains
--- the circle at is filled in, so there *is* a point
there. However, it does not attain the value -1, since the circle at
is open, meaning that the point is missing. The range
is .

A graph is * increasing* if it goes up from left
to right; it is * decreasing* if it goes down
from left to right.

* Example.* Consider the function *whose
entire graph* is shown below.

The function increases on the intervals and on . It decreases on the interval .

If f and g are functions, their * composite* is

The does *not* mean multiplication; it means that
g is "inside" of f, or that the output of g is fed into f.

* Example.* Let and let . Find , , , and .

Notice that is not the same as .

* Example.* Let and let . Find , , , and .

Notice that is not the same as .

Copyright 2008 by Bruce Ikenaga