 # 48 Different Types of Functions and their Graphs [Complete list]

There are different types of functions in Mathematics.

In the previous lesson, we have learned What is a function? Now in this chapter, we will learn about 48 Different Types of Functions Graphs.

We have tried to include all types of functions and their graphs.

## Algebraic function

### Polynomial function

A polynomial in the variable $x$ is a function that can be written in the form,

$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+…..+a_{2}x^{2}+a_{1}x+a_{0}$

where $a_{n},\: a_{n-1},…..,\: a_{2},\: a_{1},\: a_{0}$ are constants.

We call the term containing the highest power of $x\: (i.e.,a_{n}x^{n})$ the leading term, and we call $a_{n}$ the leading coefficient.

The degree of the polynomial is the power of x in the leading term.

There are different types of polynomial function based on the degree of the leading term and they are

### Power function

A Power Function is expressed as

$y=ax^{n}$,

where a is a constant and n is an integer.

Example:

• $y=x$,
• $y=x^{2}$,
• $y=x^{3}$,
• $y=x^{-1}=\frac{1}{x}$,
• $y=x^{-2}=\frac{1}{x^{2}}$,
• $y=x^{-3}=\frac{1}{x^{3}}$.

### Rational function

The quotient of two polynomials is called a Rational function.

Rational function is expressed in the form

$f(x)=\frac{g(x)}{h(x)},h(x)\neq 0$,

where g(x) and h(x) are polynomial functions.

The domain of a rational function is the set of all real numbers excepting those x for which $h(x)=0$.

Example:

• $y=\frac{1}{x^{2}}$,
• $y=\frac{x^{3}-x^{2}+1}{x^{5}+x^{3}-x+1}$.

### Irrational function

The functions that can not be expressed as a quotient of two polynomial functions are called Irrational Function.

Irrational functions involve radical, trigonometric functions, hyperbolic functions, exponential and logarithmic functions etc.

Example:

• $y=\sqrt{x^{3}}=x^{\frac{3}{2}}$,
• $y=2^{x}$,
• $y=log_{a}\: x$.

### Modulus function or Absolute value function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined $f(x)=\left | x \right |,x\epsilon \mathbb{R}$. The range of the function is {$x\epsilon \mathbb{R}:-1\leq x\leq 1$}.

f is equivalently expressed as $f(x)=\left | x \right |$

or as

f is called the Modulus function (Absolute value function).

### Signum function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined by $f(x)=sgn\: x,\: x\epsilon \mathbb{R}$

This function is called signum function and range of signum function is {-1, 0, 1}.

Signum function is equivalently expressed as

### Greatest integer function or Floor function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined by $f(x)=\left [ x \right ], x\epsilon \mathbb{R}$.

[x] is the greatest integer not greater than x (i.e., smaller than x) and the range of the function is $\mathbb{Z}$.

f is equivalently expressed as

f is called the greatest integer function or Floor function.

Example:

• $\left [ 2.3247 \right ] =2$,
• $\left [ 0.231 \right ] =0$,
• $\left [ -8.3247 \right ] =-9$,
• $\left [ -0.78 \right ] =-1$.

### Least integer function or Ceiling function

The least integer function (Ceiling function) is expressed as $y= \lceil x \rceil$.

Here $\lceil x \rceil$ is the least integer greater than x and the range of the function is $\mathbb{Z}$.

Example:

• $\lceil 2.3247 \rceil =3$,
• $\lceil 0.231 \rceil =1$,
• $\lceil -8.3247 \rceil =-8$,
• $\lceil -0.78 \rceil =-0$.

### Step function

A function $f$ defined on $I=\left [ a,b \right ]$ is said to be a step function on $I$ if there exist finite number of points $x_{0},x_{1},x_{2},…..,x_{n}\: (a=x_{0} such that $f$ is a constant on each open subinterval $\left ( x_{k-1},x_{k} \right )$ of [a,b].

That is, for each k=1, 2, ….. , n, there is a real number $s_{k}$ such that $f(x)=s_{k}$ for all $x\epsilon (x_{k-1},x_{k})$. $f(x_{k-1}), f(x_{k})$ need not be same as $s_{k},\: k=1,2,…..,n$.

### Fractional part function

Fractional part function is expressed as

y = {x} = x – [x]

or as

Example

• {1.5} = 1.5 – [1.5] = 1.5 – 1 = .5
• {-1.4} = -1.4 – [-1.4] = -1.4 – (-2) = -1.4 + 2 = 0.6,
• {-1} = -1 – [-1] = -1 + 1 = 0,
• {1} = 1 –  = 1 -1 =0.

### Dirichlet function

Let $a, b\epsilon \mathbb{R}\: \left ( a \neq b \right )$. The Dirichlet function is defined as

• The function f defined above has its domain $(-\infty ,\infty )$.
• The value of f jump in finitely often from a to b and back, in any interval of x.
• Dirichlet function has an analytic form which is $f(x)=\lim_{m\rightarrow \infty }\lim_{n\rightarrow \infty }cos^{2n}m!\pi x$.
• This function can not be represented by a graph on the Euclidean plane.

Example 1. If we take a=1 and b=0 then the dirichlet function is as follows

• Domain = $(-\infty ,\infty )$,
• The value of $f$ jump in finitely often from 1 to 0 and back, in any interval of x.

Example 2. If we take a=1 and b=-1 then the dirichlet function is as follows

• Domain = $(-\infty ,\infty )$,
• The value of $f$ jump in finitely often from 1 to -1 and back, in any interval of x.

## Transcendental function

Now we learn about some functions other than the algebraic functions.

They are called Transcendental functions and they are:

1. Trigonometric function or circular function,
2. Inverse Trigonometric function or Inverse circular function,
3. Exponential function,
4. Logarithmic function,
5. Hyperbolic function,
6. Inverse hyperbolic function

### Trigonometric function or circular function

The Trigonometric (or circular) functions are

1. y = sin x,
2. y = cos x,
3. y = tan x,

and their reciprocals are

1. y = cosec x,
2. y = sec x,
3. y = cot x.

We shall always take the radian measure of the angle as the argument (input) x i.e., the value of y = sin x at $x=x_{0}$ is equal to the sine of the angle of $x_{0}$ radians.

If $x=x_{0}=\frac{\Pi }{2}$, then $y=sin\: x=sin\: x_{0}=sin\left ( \frac{\Pi }{2} \right )=1$.

The trigonometric functions are periodic.

• Functions sin x and cos x have a period $2 \Pi$,
• Functions tan x and cot c have a period $\Pi$.

### Inverse Trigonometric function or Inverse circular function

The functions $y=sin^{-1}x$ (or Arc sin x), $y=cos^{-1}x$ (or Arc cos x), $y=tan^{-1}x$ (or Arc tan x), etc., are inverse to trigonometric functions sin x, cos x, tan x, etc. are called Inverse Trigonometric function or Inverse circular function.

There are 6 Inverse Trigonometric functions or Inverse circular functions and they are

1. inverse function of sin x is $sin^{-1}x$ or Arc sin x,
2. inverse function of cos x is $cos^{-1}x$ or Arc cos x,
3. inverse function of tan x is $tan^{-1}x$ or Arc tan x,
4. inverse function of cosec x is $cosec^{-1}x$ or Arc cosec x,
5. inverse function of sec x is $sec^{-1}x$ or Arc sec x,
6. inverse function of cot x is $cot^{-1}x$ or Arc cot x.

The values of these functions express radian measures of the angles or the lengths of the arcs of a unit circle.

i.e., if $y=sin^{-1}x=1$, then $x=sin\left ( 1 \right )=\frac{\Pi }{2}$, a radian measure.

### Exponential function

An exponential function has the form $y=a^{x}$ where $a>0\: and \: a\neq 1$.

Range of exponential function belongs to $\left ( 0,\infty \right )$.

Depending on the value of a here two case arise and they are

Case 1

When a>1, $y=a^{x}$ is strictly increasing function.

Example: $2^{x},3^{x},4^{x}$ etc.

Case 2

When 0<a<1, $y=a^{x}$ is strictly decreasing function.

Example: $\left ( \frac{1}{2} \right )^{x}, \left ( \frac{1}{3} \right )^{x}, \left ( \frac{1}{4} \right )^{x}$ etc.

When a = e, the exponential function takes the form

$y=e^{x}=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+…..$

### Logarithmic function

The function $f(x)=log_{a} \: x;\: \left ( x,a> 0 \right )$ and $a\neq 0$ is a logarithmic function.

Thus, the domain of the logarithmic function is all real positive numbers and their range is the set $\mathbb{R}$ of all real numbers.

We have seen that $y=a^{x}$ is strictly increasing when a>1 and strictly decreasing when 0<a<1.

So the function is invertible.

The inverse of this function is denoted by $log_{a} \: x$, we write

$y=a^{x}\Rightarrow x=log_{a}\: y$;

where $x\epsilon \mathbb{R}$ and $y\epsilon \left ( 0,\infty \right )$.

Writing $y=log_{a} \: x$ in place of $x=log_{a} \: y$, we have the graph of $y=log_{a} \: x$.

Thus the logarithmic function is also known as inverse of the exponential function.

### Hyperbolic function

There are 6 hyperbolic functions and they are defined by

1. hyperbolic sine of x written as $sinh \: x=\frac{1}{2}(e^{x}-e^{-x})$,
2. hyperbolic cosine of x written as $cosh \: x=\frac{1}{2}(e^{x}+e^{-x})$,
3. hyperbolic cosec of x written as $cosech \: x=\frac{2}{e^{x}-e^{-x}}$,
4. hyperbolic sec of x written as $sech \: x=\frac{2}{e^{x}+e^{-x}}$,
5. hyperbolic tan of x written as $tanh\: x=\frac{sinh\: x}{cosh\: x}=\frac{e^{x}-e^{x}}{e^{x}+e^{x}}$,
6. hyperbolic cot of x written as $coth\: x=\frac{1}{tanh\: x}=\frac{e^{x}+e^{x}}{e^{x}-e^{x}}$.

The following two results follow from our definitions:

• $cosh\: x+sinh\: x=e^{x}$,
• $cosh\: x-sinh\: x=e^{-x}$.

### Inverse hyperbolic function

The inverse of the hyperbolic function discussed above are:

1. $sinh^{-1}\: x=log\left ( x+\sqrt{x^{2}+1} \right )$ (defined for all real x),
2. $cosh^{-1}\: x=log\left ( x+\sqrt{x^{2}-1} \right )\: \left ( x\geq 1 \right )$,
3. $tanh^{-1}\: x=\frac{1}{2}log\frac{1+x}{1-x}\: ,\: ( -1< x< 1$ or $\left | x \right |< 1)$,
4. $coth^{-1}\: x=\frac{1}{2}log\frac{x+1}{x-1}\: ,\: ( \left | x \right |> 1)$,
5. $sech^{-1}\: x=log\frac{1+\sqrt{1-x^{2}}}{x},\: \left ( 0< x< 1 \right )$
6. $cosech^{-1}\: x=log\frac{1\pm \sqrt{1+x^{2}}}{x}$, (+ve sign if x > 0 and -ve sign if x < 0)

## Even and Odd function

For $a\epsilon \mathbb{R}*$, let $D$ be the symmetric interval (-a,a).

### Even function

A function $f:D\rightarrow \mathbb{R}$ is said to an even function if $f(-x)=f(x)$ for all $x\epsilon D.$

Example: The function $f:D\rightarrow \mathbb{R}$ defined by $f(x)=x^{2}, f(x)=cos\: x$ are even functions on $\mathbb{R}$.

### Odd function

A function $f:D\rightarrow \mathbb{R}$ is said to an even function if $f(-x)=-f(x)$ for all $x\epsilon D$.

Example: The function $f:D\rightarrow \mathbb{R}$ defined by $f(x)=x, \: f(x)=sin\: x, \: sgn\: x$ are odd functions on $\mathbb{R}$.

## Implicit and Explicit function

### Explicit function

If a function is directly expressed as y as a function of, then it is called an explicit function.

Example:

• $y=x+1$,
• $y^{2}=4ax$.

### Implicit function

If a function is not expressed as a function of x directly then it is called an implicit function.

Example:

• $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$,
• $y-e^{x}=0$.

## Periodic function

A function $f:D\rightarrow \mathbb{R}$ is said to be a periodic function if there exists a positive real number $p$ such that $f(x+p)=f(x),\: p$ being called the period of a function.

Equivalently, the least positive real number $p$ (if exists) is said to be the period of a function $f:D\rightarrow \mathbb{R}$ if $f(x+np)=f(x)$ holds in $D$ for all integer n.

Example: $sin\: x$ is a Periodic function of period $2\pi$.

$sin\: x=sin\: (x+2n\pi )$ for all integer n and $2\pi$ is the least positive value of $2n\pi$.

Therefore period of sine function is $2\pi$.

For better understanding watch the video given below (duration: 8 seconds).

## Inverse function

Let $f:A\rightarrow B$ be s function whose domain is A and whose range ($\equiv codomain$) is B.

Then by definition, for each $x\epsilon A$, there exist an unique $y=f(x)\epsilon B$ ;$x$ is called the argument and $y \: or \: f(x)$ is the value of the function at $x$.

If now the function is one to one, then we shall get for each $y\epsilon B$, a unique $x$ in $A$. This correspondence is called the inverse mapping or inverse function, denoted by $f^{-1}$.

We may consider that $f^{-1}$ maps each $y=f(x) \epsilon B$ to a unique $x=f^{-1}(y)\epsilon A$. Thus when $f^{-1}$ exists, $B$ is its domain and $A$ is its range.

Then we can say

$f:x\rightarrow f(x);\: \: f^{-1}:f(x)\rightarrow x$

Example:

Let the domain be $A=$ { $x\epsilon \mathbb{R} : x\geq 0$ } and $f(x)=x^{2},x\epsilon A$.

Then range $f(A)=$ { $x\epsilon \mathbb{R}:x\geq 0$ } $=E$(say). Then $f:A\rightarrow E$ is one to one as well as onto.

The inverse function $f: E \rightarrow A$ is defined by $f^{-1}(x)=\sqrt{y}=\sqrt{x^{2}}=x$ since $x>0$.

This inverse function is called the square root function.

## Restriction function

Let $D\subset \mathbb{R}$ and $f:D\rightarrow \mathbb{R}$ be a function. Let $D_{0}$ be a non empty subset of $D$. The function $g:D_{0}\rightarrow \mathbb{R}$ defined by $g(x)=f(x), x\epsilon D_{0}$ is said to be the restriction of $f$ to $D_{0}$ and $g$ is denoted by $f/D_{0}$

Example:

Let $f:D\rightarrow \mathbb{R}$ be defined by $f(x)=sgn\:x, x\epsilon \mathbb{R}$.

Let $D_{0}$={$x\epsilon \mathbb{R}:x> 0$}. Then the restriction function $f/D_{0}$ is defined by $f/D_{0}(x)=1,x>0$.

## Equal function

Let $D\subset \mathbb{R}$. The function $f:D\rightarrow \mathbb{R}$ and $g:D\rightarrow \mathbb{R}$ having the same domain are said to be equal if $f(x)=g(x)$ for all $x\epsilon D$.

Example: Let $f(x)=\left | x \right |,x> 0;\: g(x)=x,x> 0$ be two functions.

Then f and g have the same domain {$x\epsilon \mathbb{R}:x> 0$} and $f(x)=g(x)$ for all x in the domain.

Therefore $f=g$

## Monotone function

Let $I\subset \mathbb{R}$ be an interval. A function $f:\: I\rightarrow \mathbb{R}$ is said to be monotone on $I$ if $f$ is monotone increasing or monotone decreasing on $I$.

Now you are thinking what is a monotone increasing function and what is a monotone decreasing function?

There are 4 types of monotone function

1. Monotone increasing function,
2. Strictly Monotone increasing function,
3. Monotone decreasing function,
4. Strictly Monotone decreasing function.

### Monotone increasing function

A function $f:\: I\rightarrow \mathbb{R}$ is said to be monotone increasing function on $I$ if $x_{1},x_{2}\epsilon I$ and $x_{1}< x_{2}\Rightarrow f(x_{1})\leq f(x_{2})$

Example: Let $f(x) = sgn\: x, x\epsilon [-1,1]$

$x_{1} < 0, x_{2} < 0$ and $x_{1} < x_{2} \Rightarrow f(x_{1})=f(x_{2})$

$x_{1} < 0, x_{2} > 0$ and $x_{1} < x_{2} \Rightarrow f(x_{1})

$x_{1} > 0, x_{2} > 0$ and $x_{1} < x_{2} \Rightarrow f(x_{1})=f(x_{2})$

Therefore f is monotone increasing on [-1,1].

### Strictly Monotone increasing function

A function $f:\: I\rightarrow \mathbb{R}$ is said to be strictly monotone increasing function on $I$ if $x_{1},x_{2}\epsilon I$ and $x_{1}< x_{2}\Rightarrow f(x_{1})< f(x_{2})$.

Example: y=x (latex]x\epsilon \mathbb{R}[/latex]) is a strictly monotone increasing function because for every $x_{1}> x_{2}\Rightarrow f(x_{1})> f(x_{2})$.

### Monotone decreasing function

A function $f:\: I\rightarrow \mathbb{R}$ is said to be monotone increasing function on $I$ if $x_{1},x_{2}\epsilon I$ and $x_{1}< x_{2}\Rightarrow f(x_{1})\geq f(x_{2})$

### Strictly Monotone decreasing function

A function $f:\: I\rightarrow \mathbb{R}$ is said to be strictly monotone decreasing function on $I$ if $x_{1},x_{2}\epsilon I$ and $x_{1}< x_{2}\Rightarrow f(x_{1})> f(x_{2})$

Example: Let $f(x)=1-x,x\epsilon \mathbb{R}$.

$x_{1},x_{2}\epsilon \mathbb{R}$ and $x_{1} f(x_{2})$.

Therefore $f$ is strictly decreasing on $\mathbb{R}$.

## Bounded function

A real valued function $f$ defined on a domain $is said to be bounded if there exist two real numbers [latex]k$ and $K$ such that, $k\leq f(x)\leq K$.

$k$ is said to be a lower bound and $K$ an upper bound of $f$ on $D$.

If $f$ is not bounded, it is said to be unbounded.

Example: $sin\: x$ is a bounded function because $-1< sin\: x< 1$.

## Parametric function

If both dependent variable $(y)$ and the independent variable $(x)$ are expressed as a function of a third variable $t$ or $\left ( \theta \right )$, we say that the function has been represented parametrically.

This third variable $t$ or $\left ( \theta \right )$ is called a parameter.

Example: $x=at^{2}$, $y=2at$ represent parametrically $y^{2}=4ax$ (a parabola).

## Function of a function or Composite Function

Let $u=f(x)$ and $y=\phi (u)$ be two functions such that $f$ is defined over a set $S$ of real numbers and $\phi$ is defined over a set $T$ of real numbers.

Suppose every $f(x)$ for all $x\epsilon S$ is a member of $T$.

Then clearly the two relations $u=f(x)$ and $y=\phi (u)$ determine $y$ as a function of $x$ defined over $S$.

We call $y$ as a function of a function or Composite function.

Example 1. Let $u=x^{3}$, $y=sin\: u$.

Then the Composite function (Function of a function) is

$y=sin\: u=sin\: x^{3}$

i.e., $y=sin\: x^{3}$

Example 2. Let $u=x^{2}+1$ and $y=\sqrt{u}$.

Then similarly $y=\sqrt{x^{2}+1}$

We hope you understand every different types of functions and their graphs.

Still have any question on the topic different types of functions, please let us know in the comment section.

1. 