Mathematical Functions: Definition and Properties

by ms-academy | Aug 25, 2021 | Math Learning

Table of Contents

While studying mathematics you may have noticed that functions are widely used in nearly every subject. In a matter of fact, they are one of the most basic and fundamental objects in mathematics, so in this article, we will learn about functions, their definition, and their properties.

What is a function

We call a function a given relation between elements of two sets, in a way that each element of the first set is associated with one and only one element of the second set. For example, from the set of Natural Number $\mathbb{N}$ to the set Natural Numbers $\mathbb{N}$ , or from the set of Integers $\mathbb{Z}$ to the set of Real Numbers $\mathbb{R}$ .

The first set is called the Domain, it is usually denoted by $X$ and its elements by $x$ , and the second set is called the codomain noted $Y$ and its elements $y$ . A function is usually denoted by a letter like: $f$ , $g$ , or $h$ …etc.

If we take an example of a function called $f$ , and $f$ is a relation between the elements $x$ of the set $X$ and elements $y$ of the set $Y$ . $f$ is denoted by $f(x)=y$ and it reads: $f$ of $x$ equal $y$ . The element $x$ here is called the argument or the input of the function $f$ , and we call the element $y$ the value of the function or the image of $x$ by $f$ or the output of $f$ .

For example, we note:

$\begin{align*} f: & X\to Y\\ & x\longmapsto f(x) \end{align*}$

$\begin{align*} g: & \mathbb{R}\to \mathbb{R}\\ & x\longmapsto 5x^{3}+2x-4 \end{align*}$

$\begin{align*} h: & \mathbb{R^{+*}}\to \mathbb{R}\\ & h(x)= \log(x) \end{align*}$

Evaluating a function $f$ for a value of $x$ is simply the procedure of replacing $x$ by its given value and calculate the output or the value of $f(x)$ .

Graphical representation

A useful way to see the behavior of the function on its domain of definition is by drawing it graphicly on a plan with two axes on for the values of $x$ noted $X$ -axis and one for the $y$ or $f(x)$ noted $Y$ -axis. A function $f$ is represented by all the pairs $(x,f(x))$ on a plane generating the Graph of the function $G_{f}$ .

The domain of definition

As we said before, the domain of definition is the set of the possible argument (or inputs) of the function, and every element of this set has exactly one image on the codomain. For better understanding let’s see some examples of functions and their domain of definition!

Example: determine the domain of the following functions:

$F(x)=\dfrac{1}{x}; \;\; G(x)=\dfrac{1}{\sqrt{x-1}}; \;\; H(x)=\dfrac{3x^{2}-4}{x^{2}-1}$

For $F$ , we can calculate the value of $F(x)$ for every real number $x$ except for $0$ , because we cannot divide by $0$ i.e., $\dfrac{1}{0}$ , so the domain is all the real numbers except $0$ meaning $D_{F}=\mathbb{R^{*}}$ we can also note $D_{F}=]-\infty; 0[\cup]0; +\infty[$ .

Let’s take a look at the graph of F:

For the case of the second function, to determine the domain of definition of $G$ we need to determine and exclude the values of $x$ that make:

the denominator equal to zero, because we cannot divide by $0$ .
the square root of $x-1$ less than zero, because we can’t evaluate the square root of a negative number (In the space of Real Numbers $\mathbb{R}$ ).

In other terms, we can evaluate the value of $G(x)$ for every real number except for the ones that make the denominator less or equal to $0$ .

So, for the denominator to be equal to $0$ , the expression $\sqrt{x-1}$ has to be equal to $0$ , We have:

$\begin{align*} \sqrt{x-1} &=0\\ \Leftrightarrow x-1 &=0\\ \Leftrightarrow x &=1 \end{align*}$

So, the value $x=1$ doesn’t belong in the domain of definition $D_{G}$ .

For the square root to be less than zero, we evaluate the expression $\sqrt{x-1}<0$ , we have then:

$\begin{align*} \sqrt{x-1} &<0\\ \Leftrightarrow x-1 &<0\\ \Leftrightarrow x &<1 \end{align*}$

Therefore, the values $]-\infty; 1[$ don’t belong in the domain of definition $D_{G}$ .

So, by combining (the union) the two previous results, we get the excluded values are: $]-\infty; 1[\cup[1]\equiv]-\infty;1]$ .

We conclude that the domain of definition of $G$ is $D_{G}=]1;+\infty[$ .

(We can also take the opposite direction and instead of looking for the values to exclude we look for the possible values of $x$ , meaning the value of $x$ that make $\sqrt{x-1}\geq$ ).

Here is the graphical representation of $G$ :

For the third function $H$ , we notice that it is in the form of a fraction, so, we need to make sure that the denominator is not equal to zero and exclude the values of $x$ for which it does. In this case, we have the denominator is a polynomial of the second degree, so we need to determine its roots, the denominator $x^{2}-1$ is a remarkable identity (subtraction of two squares), meaning it may be written in the factorial form $(x-1)(x+1)$ . Now to determine the roots of the denominator all we need to do is to solve the equation:

$(x-1)(x+1)$

and we have then: $x=1$ or $x=-1$ , these two values of $x$ are excluded from the domain of definition $D_{H}$ .

Therefore, the Domain of definition of $H$ is: $D_{H}=]-\infty;-1[\cup]-1;1[\cup]1;+\infty[$ .

Let’s see the graph of $H$ :

The domain of values (or codomain)

Definition: The codomain or the set of destination of a function $f$ is the set containing all the output or image of $f$ i.e. the set containing all $f(x)$ for all $x$ in the domain. In other terms, the codomain of a function $f$ is the set of all possible outputs of $f$ .

Note that the codomain can be bigger, smaller, or entirely different from the domain.

For example, A function can have its domain as Natural numbers $\mathbb{N}$ , and the codomain as the odd natural numbers; Or it can have its domain as the natural numbers $\mathbb{N}$ and the codomain as negative integers $\mathbb{Z^{-}}$ ; Or having the domain as the set of real numbers $\mathbb{R}$ and the same for the codomain.

Let’s take look at an interesting example:

Is the Square root a function? If we take $g(x)$ as the square root of a positive real number $x$ , so we have $g(4)=2$ or $-2$ , meaning the $g(x)$ is defined as follows

$g(x)=\left\{\begin{array}{lr} \sqrt{x}\\ -\sqrt{x} \end{array}\right.$

then $g(x)$ is not a function, because as we mentioned at the beginning every element of the domain set must have one and only one image in the codomain, but in this case, each element of the domain has two images one positive and one negative, and therefore $g(x)$ is not a function. This can be fixed by limiting the codomain to only positive real numbers!

$g:\mathbb{R^{+}} \to \mathbb{R^{+}}$

Also keep in mind, the Radical sign $\sqrt{}$ which we use when calculating the root square of a number always means the positive square root, that’s why $y=f(x)=\sqrt{x}$ is a function and always givesa positive real numbers as an output.

$\begin{align*} f: \mathbb{R^{+}} \to \mathbb{R^{+}}\\ f(x)= \sqrt{x} \end{align*}$

Here is a graphical representation:

Note that the codomain may contain elements that are not images of any element of the domain, in other terms an element $y$ in the codomain doesn’t necessarily mean that there is an element $x$ in the domain that if we take $x$ as input, we will get $y$ as an output. For better understanding let’s take an example: If we take $f$ as follow:

$\begin{align*} f: & \mathbb{R}\to \mathbb{R}\\ f(x)&=x^{2}-3 \end{align*}$

We have here the domain of $f$ is the set of real numbers $\mathbb{R}$ , and the codomain is the set of real numbers $\mathbb{R}$ . but the image of $\mathbb{R}$ by $f$ (or the inputs) is not all $\mathbb{R}$ , it is a sub-set of the codomain $\mathbb{R}$ . In a matter of fact, The image of $\mathbb{R}$ by $f$ is $[-3;+\infty[$ and this is what we call the Range of $f$ , Therefore the image of every element in the domain is contained in the codomain, but not necessarily the opposite.

For more clarification let’s take a look at the graph of $f$ and see its range.

Notice that the graph doesn’t go beyond $-3$ on the $Y$ -axis which means the range of $f$ is $[-3;+\infty[$ and the range is a sub-set of the codomain $\mathbb{R}$ since we have: $f:\mathbb{R}\to\mathbb{R}$ .

Even and odd functions

Even function

Definition: a function $f$ is called an even function, if and only if it verifies the following property:

$f(x)=f(-x)$

Or equivalently

$f(x)-f(-x)=0$

Note that first, the function must have $x$ and $-x$ as elements of its domain which means the domain must be symmetrical in the first place.

Graphicly speaking, the graph of an even function is symmetrical with respect to the $Y$ -axis.

Let’s take a look at some examples of even functions with their respective graphs:

The absolute value: $f(x)=\vert x \vert$

The cosine: $g(x)=\cos(x)$

The function: $h(x)=(x^{2}-2)^{2}-3$

Odd function

Definition: a function $f$ is an odd function if and only if it verifies the following:

$f(-x)=-f(x)$

Or equivalently

$f(x)+f(-x)=0$

Note that first, the function must have $x$ and $-x$ as elements of its domain which means the domain must be symmetrical in the first place.

Graphically speaking, the graph of an odd function is symmetrical about the origin $O(0,0)$ .

Let’s take a look at some examples of odd functions and their respective graphs:

The identity: $f(x)=x$

The sine: $g(x)=\sin(x)$

The function: $h(x)=x^{5}-4x^{3}+x$

Some properties of Even and odd functions

The absolute value of an odd function is an even one (the most obvious example, the identity function $f(x)=x$ is odd and the absolute value of it $|f(x)|=|x|$ is even).
The summation of two odd functions is an odd one.
The summation of two even functions is an even one.
The subtraction of two odd functions is an odd one.
The subtraction of two even functions is an even one.
The multiplication of two even functions is an even one.
The multiplication of two odd functions is an even one.
The multiplication of an odd function with an even function is an odd one.
The division of two even functions is an even one.
The division of two odd functions is an even one.
The division of an even function and an odd function is an odd one.

You want to try out these properties! Check out the window below and see the results of the summation, subtraction, multiplication, or division of even and odd functions. Check the operation you want to display (one or multiple), and watch the respective graphs. Have fun!!!!!

Periodic functions

We call a periodic function a function that its values (outputs) repeat regularly.

Definition:

A function $f$ is called periodic with a period $T$ different than $0$ , if and only if for any $x$ , $x+T$ , and $x-T$ of its domain, it verifies the following:

$f(x+T)=f(x-T)=f(x)$

Where $T$ is called the period.

Graphicly speaking, a periodic function has a part of its graph repeats regularly.

Let’s take a look at some examples of periodic functions and their respective graphs.

The sine: $f(x)=\sin(x)$

The cosine: $g(x)=\cos(x)$

The tangent: $p(x) \tan(x)$

Some properties of periodicity

If $f$ is periodic with a period $T$ , then $k.T$ , where $k\in\mathbb{N}$ , is also a period for $f$ .
To graph a periodic function with a period $T$ , all we need to do is to graph an interval of length $T$ , and then duplicate it at both ends left and right, or in other terms shift the graphed section along the $X$ -axis at a distance $nT$ ( $n\in\mathbb{N}$ ) left and right.
If $f$ is periodic with a period $T$ , then for all $x$ in the domain of definition of $f$ , and for all natural number $n$ we have:
$f(x)=f(x+nT)$

.
If $f$ is periodic with a period $T$ , then $cf(x)$ , $f(cx)$ , and $f(x)+c$ are periodic with period $T$ , where $c$ is a real number diffrent than $0$ .

Conclusion

In this article we learned a little bit about functions and some of their properties and characteristics, this may be considered as an introduction to how to study a function and detect its properties which will help for better understanding of its behavior, but keep in mind this is just an introduction and there is a lot more to learn! isn’t that amazing! So who is eager for another post to learn more about functions!!!!!

Are you looking for another interesting article! Take a look at how to solve polynomial functions of first, second, and third degree. Enjoy!!!

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