Briefly, a function in mathematics is a procedure that operates on numbers (or possibly other mathematical objects) to return other numbers (or possibly mathematical objects). For instance, the procedure could just double the number given to it, or it could add 4 to it, or even do nothing to it at all. When a function operates only on numbers as opposed to other mathemtical oobjects and returns only numbers, it is called a real-valued function of a real variable.

We obtained the graph of the numerically specified function above by plotting points with the values of $f(x)$ used as $y$-coordinates. So, the points we plotted had the form $(x, y) = (x, f(x))$. Regardless of how a function is specified, we obtain its graph in the same way: As we saw above (see "Functions and domains" at the top of the page) the domain of a function, if not specified, is taken to be its natural domain, but in many applications we may want to restrict it to a range of values corresponding to a real situation. For instance, if $C(x) = x^2+5$ represents the cost of produce $x$ kilograms of ectoplasm essence in a day, the function has no real meaning if $x$ is negative, so we should really restrict the domain to $[0,\infty)$, or, if the formula does not apply in the case when no ectoplasm essence is made, to $(0,\infty)$. Further, if it is not possible to make more than 20 kilograms in a day, then we should further restrict the domain to $[0, 20]$ or $(0,20]$.

We represent these possibilities graphically as follows:

A solid dot at an endpoint in a graph indicates that its $x$ value is in the domain, so the point is actually a point on the graph. An open dot indicates that its $x$ value is in not in the domain, so the point is actually deleted from the graph. Sometimes we need more than a single formula to specify a function algebraically, as in the following example, similar to Example 2 in the textbook:

The number, in millions, of Facebook members from 2004 to 2009 can be approximated by the following function ($t=0$ represents the start of 2004):^†
†#[Source for data][Fuente de datos]#: http://www.facebook.com

• We use the first formula: $\color{green}{4t}$ (see the green portion of the graph) to calculate $n(t)$ if $0 \leq t \leq 3$, or, equivalently, $t$ is in $[0, 3]$.
• We use the second formula: $\color{coral}{50t-138}$ (see the orange portion of the graph) to calculate $n(t)$ if $3 \lt t \leq 5$, or, equivalently, $t$ is in $(3, 5]$.

Thus, for instance,

$n(2.5) = \color{green}{4(2.5)} = 10 \qquad$	We use the first formula because $0 \leq 2.5 \leq 3$.
	Membership midway though 2006 ($t=2.5$) was 10 million.
$n(3) = \color{green}{4(3)} = 12 \qquad$	We use the first formula because $0 \leq 3 \leq 3$.
	Membership qt the start of 2007 ($t=3$) was 12 million.
$n(3.5) = \color{coral}{50(3.5)-138} = 37 \qquad$	We use the second formula because $03 \lt 3.5 \leq 5$.
	Membership midway though 2007 ($t=3.5$) was 37 million.

To end this tutorial, a short note on different ways of writing down an algebraically defined function. Normally, to specify a function algebraically, we need to write down a defining equation, as in, say,

$f(x) = 3x - 2.$

If we replace the "$f(x)$" by "$y$", we get an equation with no explicit mention of any function:

$y = 3x - 2. \qquad$ #[An equation in two variables:][Una ecuación en dos variabvles]# $x$ %%and $y$

Technically, $y = 3x - 2$ is an equation and not a function. However, an equation of this type, $y = \text{expression in} x,$ can be thought of as "specifying $y$ as a function of $x$" as follows: Given any value $x,$ we obtain the value of the function at $x$ by calculating the corresponding value of $y$ in the equation. So the value of the function at $x = 1$ is just

$y = 3(1) - 2 = 1,$

#[which is the same as $f(1)$ for our original function][que es lo mismo que $f(1)$ para nuestra función original]# $f(x) = 3x - 2.$