Tutorial: Functions from the numerical, algebraic, and graphical viewpoints
(This topic is also in Section 1.1 in Finite Mathematics and Applied Calculus) #[I don't like this new tutorial. Take me back to the old tutorial!][No me gusta este nueve tutorial. ¡Regresame al tutorial más viejo!]#Resources
Basics
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 realvalued function of a real variable.
Functions and domains
A realvalued function $f$ of a real variable is a rule that assigns to each real number $x$ in a specified set of numbers, called the domain of $f,$ a single real number $f(x),$ read '$f$ of $x.$' The quantity $x$ is called the argument of $f$ and $f(x)$ is called the value of $f$ at $x.$ A function is usually specified
The domain of a function is not always specified explicitly; if no domain is specified for the function $f,$ we take the domain to be the largest set of numbers $x$ for which $f(x)$ makes sense. This 'largest possible domain' is sometimes called the natural domain.
A realvalued function $f$ of a real variable is a rule that assigns to each real number $x$ in a specified set of numbers, called the domain of $f,$ a single real number $f(x),$ read '$f$ of $x.$' The quantity $x$ is called the argument of $f$ and $f(x)$ is called the value of $f$ at $x.$ A function is usually specified
 numerically using a table of values,
graphically using a graph, or
algebraically using a formula,
The domain of a function is not always specified explicitly; if no domain is specified for the function $f,$ we take the domain to be the largest set of numbers $x$ for which $f(x)$ makes sense. This 'largest possible domain' is sometimes called the natural domain.
Examples
A function specified numerically Graph of a numerically specified function A function specified graphically A function specified algebraically
Graphing functions
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:
The graph of a function
The graph of a function consists of all possible points of the form $(x, f(x))$, for $x$ in the domain of $f$. In practice we cannot plot all these points, as there are infinitely many, so we pick a few to plot, and then "connect the dots" and hope for the best.
The graph of a function consists of all possible points of the form $(x, f(x))$, for $x$ in the domain of $f$. In practice we cannot plot all these points, as there are infinitely many, so we pick a few to plot, and then "connect the dots" and hope for the best.
Examples
%%Let $f(x) = x^2$. To draw the graph of $f$, first choose some convenient values of $x$ in the domain and compute the corresponding $y$coordinates $f(x)$:
A compendium of functions
Restricting the domain of a function
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.
#[Domain][Dominio]#: $[0,\infty)$
#[$0$ in the domain][$0$ incido en el dominio]#
#[Domain][Dominio]#: $(0,\infty)$
#[$0$ not in the domain][$0$ no en el dominio]#
#[Domain][Dominio]#: $[0,20]$
#[$0$ in the domain][$0$ incido en el dominio]#>
#[Domain][Dominio]#: $(0,20]$
#[$0$ not in the domain][$0$ no incluido en el dominio]#
Piecewise defined functions
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
$\displaystyle n(t) = \begin{cases} \color{green}{4t} & \text{if } 0 \leq t \leq 3 \\\color{coral}{50t138}& \text{if } 3 \lt t \leq 5 \end{cases}$ 
 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: $\color{coral}{50t138}$ (see the orange portion of the graph) to calculate $n(t)$ if $3 \lt t \leq 5$, or, equivalently, $t$ is in $(3, 5]$.
$n(2.5) = \color{green}{4(2.5)} = 10 \qquad$  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. 
Functions and equations
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,
$f(x) = 3x  2.$

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

$y = 3(1)  2 = 1,$
Now try the exercises in Section 1.1 in Finite Mathematics and Applied Calculus.
Copyright © 2018 Stefan Waner and Steven R. Costenoble