 # 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.