Tutorial: Functions from the numerical, algebraic, and graphical viewpoints

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Resources

Function evaluator and grapher

Excel grapher

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 real-valued function of a real variable.

Functions and domains
A real-valued 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

graphically

algebraically

and also in other ways.

Note on domains
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.

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 $f(x)$:

The following figures show the plotted points and then the curve they suggest, which is the graph of $f$:

This particular curve is called a parabola, and its lowest point, at the origin, is called its vertex.

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

$\displaystyle n(t) = \begin{cases} \color{green}{4t} & \text{if } 0 \leq t \leq 3 \\\color{coral}{50t-138}& \text{if } 3 \lt t \leq 5 \end{cases}$

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: $\color{coral}{50t-138}$ (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.

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,

$f(x) = 3x - 2.$ If we replace the "$f(x)$" by "$y$", we get an equation with no explicit mention of any function:

#[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]# $f(x) = 3x - 2.$

Now try the exercises in Section 1.1 in Finite Mathematics and Applied Calculus.