Special functions and their graphs: Have you seen a snack vending machine? When you insert a certain amount of money and press the button, a specific type of snack is pushed out for you to enjoy. Functions, similar to machines, are anything that takes one input, transforms it, and provides an output. Evidently, there are three parts: input, rules (of transformation), and output. Knowing any two values will help us calculate the third.

There are several applications of functions around us. They are the basic idea required to build a machine, keep aeroplanes flying in the air, predict natural disasters, and decide what dosage of medicine is advisable to treat a certain condition. In this article, let us discuss some special functions and their graphical representations.

What are Functions?

Function is a special kind of relation in Mathematics. With respect to relation, a function can be defined as a relation from \(A \to B\) such that all the elements in \(A\) have an image in \(B.\) In simpler words, a function is when every input in a relation has only one output.

Function	Not a function
Every input has one output.	One input has more than one output.

How to Graph Functions?

A function \(f\) can be graphed if all the ordered pairs in the plane are written as \(\left( {x,f\left( x \right)} \right)\) or if the function is defined as an equation \(y = f\left( x \right).\) Hence, the graph of a function is a special type if the function is an equation.

How To Identify Graph of Functions?

Which of these do you think are graphs of functions?

Although each of these graphs can be written as equations, not all of these are graphs of functions. To identify if a graph is of a function or not, we can do the vertical line test.

Vertical Line Test

A graph is formed by joining several points on a plane. Such a graph is a graph of a function if and only if no two points lie on the same vertical line.

Observe the first and third graphs. There are vertical line(s) that intersect the graph at more than one point. But in the case of the second graph, there are no vertical lines that intersect the graph at more than one point. Hence, we can say that only the second graph is a function graph.

Nature of Functions

The graph of functions can be increasing or decreasing in nature.

Case 1: \(y = 3x + 2.\)

Comparing this with the general form \(y = mx + c,\) we can say that the slope of the line is \(m = 3,\) and the \(y\)-intercept \(c = 2.\)
Case 2: \(y = – \frac{3}{4}x + 7.\)

Here, the slope \(m = – \frac{3}{4}\) and \(y\)-intercept \(c = 7.\)
By observing these two graphs, we can easily see that case \(1\) is an increasing function, and case \(2\) is a decreasing function. But can we identify the nature of the graph just by looking at the function?
The answer is yes! Here we use the slope \(m.\)
1.\(m\) is positive \( \to \) Increasing function
2. \(m\) is negative \( \to \) Decreasing function
For a function \(f\left( x \right)\) is said to be increasing on an interval \(I,\) if \(f\left( {{x_1}} \right) < f\left( {{x_2}} \right)\) when \({x_1} < {x_2}\) on \(I.\)
Similarly, it is said to be decreasing on \(I,\) if \(f\left( {{x_1}} \right) > f\left( {{x_2}} \right)\) when \({x_1} > {x_2}\) on \(I.\)

Increasing and Decreasing Functions

While there are increasing or decreasing functions, there are also functions that are both increasing and decreasing.

For example, observe the graph of a function \(y = f\left( x \right)\) given below.

It is evident that this function is both increasing and decreasing. It can be described as follows.

1. \(f\) is an increasing function on \(\left[ {a,b} \right]\) and \(\left[ {c,d} \right]\)
2. \(f\) is a decreasing function on \(\left[ {b,c} \right]\)

The points at which the graph changes its nature are called the turning points or relative extreme values.

Let us now study some key functions and their graphs.

List of Functions

Some functions that are defined for real numbers are as listed below.

• Identity function
• Constant function
• Polynomial function
• Rational function
• Modulus function
• Signum function
• Greatest integer function

Identity Function

The function whose output is the same as the input is called the identity function. It is the real-valued function \(f:R \to R\) defined as
\(f\left( x \right) = x,\forall x \in R\)
Here, the domain and range of \(f\) is \(R.\) Hence, we can say that every element of set \(R\) has an image on itself. The graph of \(y = x\) is shown below.

Observe that it is a linear function that passes through the origin.

Constant Function

Any function that has a slope \(m = 0,\) is called a constant function. A function \(f:R \to R\) is a constant function if,
\(f\left( x \right) = c,\forall x \in R\)
where, \(c\) is a constant. While the domain of a constant function is \(R,\) its range is \(\left\{ c \right\}.\)
Example: The graph of \(y = 4\) is a line as shown below.

Observe that the graph is parallel to the \(x\)-axis.

Polynomial Function

We know that linear functions are called identity functions. Similarly, equations that are quadratic, cubic, quartic and so on are called polynomial functions. They involve only non-negative integer exponents of \(x.\)
The standard form of a polynomial function is defined as:

\(f\left( x \right) = {a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} + {a_{n – 2}}{x^{n – 2}} + {a_{n – 3}}{x^{n – 3}} + \cdots + {a_2}{x^2} + {a_1}x + {a_0},\forall x \in R\)
Where,
\(n \to \) is a positive number and is called the degree of that polynomial
\({a_0} \to \) is a constant and is called the leading coefficient
\({a_0},{a_1},{a_2}, \ldots {a_n} \in \,R\)
Example: The graph of the polynomial \(y = {x^4} + 3{x^3} – 9{x^2} – 23x – 12\) is as shown below.

Here, notice that the graph is not a straight line, and it has more than one turning point or relative extreme value.

Rational Function

Rational numbers are those that are the ratio of two other numbers. Similarly, a rational function is that which is a ratio of two functions. A rational function is of the form,
\(f\left( x \right) = \frac{{g\left( x \right)}}{{h\left( x \right)}},h\left( x \right) \ne 0\)
The graph of the function \(y = \frac{1}{x}\) is given below.
Here, observe that the curves, though rapidly approach the \(x\)-axis and \(y\)-axis, never intersects or touches the axis. In such a condition, the line that approaches the curve but never intersects is called an asymptote.

Rational functions may have three types of asymptotes.

• Horizontal asymptotes
• Vertical asymptotes
• Oblique asymptotes

Modulus Function

Modulus function is a real-valued function \(f:R \to R\) defined by
\(f\left( x \right) = \left| x \right|,\,\forall x \in R\)
The graph of \(y = \left| x \right|\) is given below.
Observe that the graph of \(y = \left| x \right|\) is symmetrical about the \(y\)-axis. Here, \(y\) is positive for all integer values of \(x.\)
Mathematically this can be represented as,
\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {x,x \geqslant 0} \\ { – x,x < 0} \end{array}} \right.\)
Hence, the domain is \(R,\) and the range is \(\left( {0,\infty } \right).\)

Signum Function

Signum is a function that helps identify the sign of a real value function. For positive input values of the function, it attributes a \( + 1,\) and for negative input values, the function attributes a \( – 1.\) Lastly, when the input value is zero, the output value is also zero. Mathematically, the signum function can be interpreted as,
\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} { + 1,x > 0} \\ {0,x = 0} \\ { – 1,x < 1} \end{array}} \right.\)
This can also be represented as,
\(f\left( x \right) = \left\{ \begin{gathered} \frac{{|x|}}{x},x \ne 0 \hfill \\ 0,x = 0 \hfill \\ \end{gathered} \right.\)
Evidently, the domain of a signum function is \(R,\) and the range is \(\left\{ { – 1,0,1} \right\}.\) The graph of this function is shown below.

Greatest Integer Function

The function \(f:R \to R\) denoted by \(f\left( x \right) = \left| x \right|\) and can be defined as the greatest integer that is less than or equal to \(x.\) This is represented as,
\(\left| x \right| = \max \left\{ {m \in Z\mid m \leqslant x} \right\}\)
While the domain of the function is \(R,\) the range is \(Z.\) The graph of the greatest integer function is shown below.

• This function is called a step function.
• Since the greatest integer function rounds up the input number to the nearest integer less than or equal to the input number, it is also called the floor function.

Solved Examples

Q.1 Graph \(y = \left| {x + 2} \right|.\)
Sol: Given: \(y = \left| {x + 2} \right|.\)
Let’s tabulate and find some ordered pairs for the given function.

\(x\)	\( – 6\)	\( – 4\)	\( – 2\)	\(0\)	\(1\)	\(3\)
\(y = \left| {x + 2} \right|\)	\( – 4\)	\( – 2\)	\( 0\)	\(2\)	\(3\)	\(5\)
\(\left( {x,y} \right)\)	\(\left( { – 6, – 4} \right)\)	\(\left( { – 4, – 2} \right)\)	\(\left( { – 2,0} \right)\)	\(\left( {0,2} \right)\)	\(\left( {1,3} \right)\)	\(\left( {3,5} \right)\)

We can graph the function \(y = \left| {x + 2} \right|\) using these ordered pairs.

Q.2. Is \(f:{Z_5} \to {Z_5},\,f\left( x \right) = {x^5}\)an identity function?
Sol: Given \(f\left( x \right) = {x^5}\) mod \(5\)
Evaluate \(f\left( x \right) = {x^5}\) mod \(5\) for all elements \({Z_5} = \left\{ {0,1,2,3,4} \right\}.\)
\(f\left( 0 \right) = {0^5}\) mod \(5 = 0\) mod \(5 = 0\)
\(f\left( 1 \right) = {1^5}\) mod \(5 = 1\) mod \(5 = 1\)
\(f\left( 2 \right) = {2^5}\) mod \(5 = 32\) mod \(5 = 2\)
\(f\left( 3 \right) = {3^5}\) mod \(5 = 243\) mod \(5 = 3\)
\(f\left( 4 \right) = {4^5}\) mod \(5 = 1024\) mod \(5 = 4\)
Observe that the argument (input) and output are equal.
\(\therefore f\left( x \right) = {x^5}\) mod \(5\) is an identity function.

Q.3. Evaluate the function \(f\left( x \right) = \left| {\frac{1}{2}x} \right|\) using the given values.

\(-3.2\)

\(-2.32\)

\(0.78\)

\(-0.14\)

\(0\)

\(0.52\)

\(1\)

\(2.36\)

Sol:

\(x\)	\(\frac{1}{2}x\)	\(f\left( x \right) = \left[ {\frac{1}{2}x} \right]\)
\( – 3.2\)	\( – 1.6\)	\( – 2\)
\( – 2.32\)	\( – 1.18\)	\( – 2\)
\( – 0.78\)	\( – 0.39\)	\( – 1\)
\( – 0.14\)	\( – 0.07\)	\( – 1\)
\( 0\)	\( 0\)	\( 0\)
\( 0.52\)	\( 0.26\)	\( 0\)
\( 1\)	\( 0.5\)	\( 0\)
\( 2.36\)	\( 1.68\)	\( 1\)

This can be represented graphically as,

Observe that the \(\frac{1}{2}\) in the greatest integer function has stretches each piece to twice its length.

Q.4. Identify which of these is a constant function, and write the corresponding equation.

Sol: We know that the output is always the same for a constant function for any argument. This means that the graph of the identity function has no slope. It is a straight line that is parallel to the \(x\)-axis.
Hence the constant functions in these graphs are \(y = 2\) and \(y = – 1.\)

Q.5. What is the domain and range of the real-valued function \(f:R – \left\{ 0 \right\} \to R\) defined by \(f\left( x \right) = \frac{1}{x},x \in R – \left\{ 0 \right\}\)
Sol: Given: \(f\left( x \right) = \frac{1}{x}\)
Substituting some random values for \(x,\)

\(x\)	\( – 2\)	\( – 1.5\)	\( – 1\)	\( – 0.5\)	\(0.5\)	\( 1\)	\( 1.5\)	\( 2\)
\(f\left( x \right) = \frac{1}{x}\)	\( – 0.5\)	\( – 0.67\)	\( – 1\)	\( -2\)	\( 2\)	\( 1\)	\( 0.67\)	\( 0.5\)

Hence, we can say that the domain and range are all real numbers except zero. It is represented mathematically as,
\(x \in R – \left\{ 0 \right\}\)
The graph of the rational function \(f\left( x \right) = \frac{1}{x}\) is as shown below.

Summary

A function is a rule that takes an input called an argument, transforms it, and gives an output. They are relations except that functions have only one output for every argument (input). This article defines and illustrates the graphs of special functions such as identity function, constant function, polynomial function, rational function, modulus function, signum function, and greatest integer function. Each of these functions is described with their domain and range listed and sample graphs. The article also differentiates increasing and decreasing functions and suggests ways to identify them.

Frequently Asked Questions (FAQs)

Q.1. What are the special types of functions?
Ans: The common types special functions are:
1. Identity function
2. Constant function
3. Polynomial function
4. Rational function
5. Modulus function
6. Signum function
7. Greatest integer function

Q.2. What are the special features of a graph?
Ans: Any and all graphs have three features:
1. \(x\)-axis
2. \(y\)-axis
3. Origin

Q.3. What do you mean by special functions?
Ans: Functions that are well established, have specific names, and notations are called special functions.

Q.4. What are functions?
Ans: Functions are special kinds of relations where every input has only one output.

Q.5.What is a function graph in Mathematics?
Ans: To identify a function graph, we do the vertical line test. When a vertical line does not intersect a graph at more than one point, it is called a function graph.

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