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

Degree of the polynomial | Name of the polynomial function | Example |
---|---|---|

0 | Constant function | y=6 |

1 | Linear function | y=2x-1 |

2 | Quadratic function | y=x^{2}+7x-11 |

3 | Cubic function | y=x^{3}-4x^{2}-2x+3 |

4 | Quartic function | y=x^{4}+5x^{3}-8x^{2}-9x+3 |

5 | Quintic function | y=x^{5}+2x^{4}-6x^{3}+5x^{2}-13x+17 |

### 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}<x_{1}<x_{2}<x_{2}< …..<x_{n-1}<x_{n}=b) 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 -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:

- Trigonometric function or circular function,
- Inverse Trigonometric function or Inverse circular function,
- Exponential function,
- Logarithmic function,
- Hyperbolic function,
- Inverse hyperbolic function

### Trigonometric function or circular function

The Trigonometric (or circular) functions are

- y = sin x,
- y = cos x,
- y = tan x,

and their reciprocals are

- y = cosec x,
- y = sec x,
- 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

- inverse function of sin x is sin^{-1}x or Arc sin x,
- inverse function of cos x is cos^{-1}x or Arc cos x,
- inverse function of tan x is tan^{-1}x or Arc tan x,
- inverse function of cosec x is cosec^{-1}x or Arc cosec x,
- inverse function of sec x is sec^{-1}x or Arc sec x,
- 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

- hyperbolic sine of x written as sinh \: x=\frac{1}{2}(e^{x}-e^{-x}),
- hyperbolic cosine of x written as cosh \: x=\frac{1}{2}(e^{x}+e^{-x}),
- hyperbolic cosec of x written as cosech \: x=\frac{2}{e^{x}-e^{-x}},
- hyperbolic sec of x written as sech \: x=\frac{2}{e^{x}+e^{-x}},
- hyperbolic tan of x written as tanh\: x=\frac{sinh\: x}{cosh\: x}=\frac{e^{x}-e^{x}}{e^{x}+e^{x}},
- 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:

- sinh^{-1}\: x=log\left ( x+\sqrt{x^{2}+1} \right ) (defined for all real x),
- cosh^{-1}\: x=log\left ( x+\sqrt{x^{2}-1} \right )\: \left ( x\geq 1 \right ),
- tanh^{-1}\: x=\frac{1}{2}log\frac{1+x}{1-x}\: ,\: ( -1< x< 1 or \left | x \right |< 1),
- coth^{-1}\: x=\frac{1}{2}log\frac{x+1}{x-1}\: ,\: ( \left | x \right |> 1),
- sech^{-1}\: x=log\frac{1+\sqrt{1-x^{2}}}{x},\: \left ( 0< x< 1 \right )
- 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 .

Read more: Difference between implicit and explicit function

## 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?

Then keep reading.

There are 4 types of monotone function

- Monotone increasing function,
- Strictly Monotone increasing function,
- Monotone decreasing function,
- 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})<f(x_{2})

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.

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Additionally, you can read:

Please add the formulas of the nth roots to find simply the zeros or some polynomial function like cubic,quartic,quantic,hexic and soon.