# Functions and Graphs

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 .

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