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Functions-Types of Function

July 20, 2022
written by Azhar Ejaz

What is Function in Math?

A function is a rule that relates each element of a set X with exactly one element of set Y. It also is defined as the relationship between one variable (the independent variable) and another variable (the dependent variable).

  • “…each element…” means that every element in set X is associated with or connected to at least one element in set Y.

(Note that there may be some elements in Y that remain unrelated to any elements in X, and that’s acceptable.)

  • “…exactly one…” means that a function is single-valued means that there is only one output of X.

The formula for the area of a circle, A = πr2, gives the dependent variable A (the area) as a function of the independent variable r (the radius).

 Functions are fundamental building blocks in mathematics, and they have various applications in science, engineering, and computer science.

Representation of Functions

Let f be a function from X to Y is represented as                       

f :X Y                    

read as “y is a function of x”.

Functions are generally represented as y = f(x).

Let, f(x) = x3.

It is said as f of x is equal to x cube.

Functions can also be represented by g(), t(),… etc.

The domain of a Function

The domain of a function is the set of all possible inputs for the function.

The codomain of a Function

The codomain or set of destinations of a function is the set into which all of the output of the function is constrained to fall.

Range of a Function

The range of a function refers to all the possible outputs of a function.


Let f(x) = x2. Find the domain and range of f.


                  f(x) is defined for every real number x. Further for every real number x, f(x) = x2 is a non-negative real number.

 So, Domain f = Set of all real numbers.

 Range f = Set of all non-negative real numbers.

Vertical line test

A vertical line test is used to determine whether a curve is a function or not. If any curve cuts a vertical line at more than one point, then the curve is not a function.

Types of Function

There are various types of functions given as:

  • One-to-one Function (Injective Function)
  • Onto Function (Surjective Function)
  • One to One and Onto function (bijective function)
  • Into Function
  • Polynomial Function
  • Linear Function
  • Constant Function
  • Identity Function
  • Quadratic Function
  • Rational Function
  • Algebraic Functions
  • Cubic Function
  • Even and Odd Function
  • Periodic Function
  • Composite Function

Into Function

If a function f: A → B  is such that Ran f ⊂ B i.e., Ran f ≠ B, then f is said to be a function from A into B.

Fig f is a function. But Ran f ≠ B.

Therefore, f is a function from A to B.

image showing the into function

 Onto (Surjective) function

If a function f: A→B is such that

Ran f = B i.e., every element of B is the image of some elements of A, then f is called an onto function or a surjective function.

image showing the onto function

One to One and Into (Injective) function

 If a function f from A into B is such that second elements of no two of its ordered pairs are equal, then it is called an injective (1 – 1, and into) function. The function shown in Fig is such a function.

image showing the One to One and Into (Injective) function

One to One and Onto function (bijective function)

 If f is a function from A onto B such that the second element of no two of its ordered pairs are the same, then ƒ is said to be (1-1) function from A onto B. Such a function is also called a (1-1) Correspondence between A and B. It is

also called a bijective function. Fig shows a (1-1) correspondence between the sets A and B.

(a, z), (b, x) and (c, y) are the pairs of corresponding elements i.e., in this case

f= {(a, z), (b, x). (c, y)} which is a bijective function or (1-1) correspondence

between the sets A and B.

image showing the One to One  and Onto function (bijective function)

Polynomial Function

A polynomial function is a function that is defined by a polynomial expression in one or more variables. It consists of non-negative integer powers of the variable(s) multiplied by constants.

Example: f(x) = 3x^2 + 2x – 1 is a polynomial function of degree 2.

Linear Function

A linear function is a polynomial function of degree 1, and its graph is a straight line. It has the form f(x) = mx + b, where m is the slope and b is the y-intercept.

Example: f(x) = 2x + 3 is a linear function.

Constant Function

A constant function is a function that always returns the same constant value, regardless of the input.

Example: f(x) = 5 is a constant function as it always gives the output 5.

Identity Function

The identity function is a function that returns the same value as its input. The domain and codomain are the same for the identity function.

Example: f(x) = x is the identity function.

Quadratic Function

A quadratic function is a polynomial function of degree 2. Its graph is a parabola.

Example: f(x) = x^2 + 2x – 3 is a quadratic function.


  A rough sketch of the functions

i) ((x, y) | 3x+y = 2)

ii) ((x, y) l y= ½ x2)


i) The equation defining the function is 3x + y = 2

⇒ y = -3x + 2

We know that this equation, being linear, represents a straight line. Therefore, for drawing its sketch or graph only two of its points are sufficient.


x=0, y = 2.

y = 0, x = 2/3 = 0.6 nearly.

 So two points on the line are A (0, 2) and B = (0.6, 0).

Joining A and B and producing the AB bar in both

directions, we obtain the line AB i.e., the graph of the given function.

image showing the Linear and Quadratic Functions

ii) The equation defining the function is

y =1/2 x2.

image showing the parabola curve of function

Corresponding to the values {…,-3, -2, -1, 0, 1, +2, +3} of x, values of y are 0, .5, 2, 4.5, …

We plot the points

(0, 0), (± 1, .5), (± 2, 2),(± 3, 4.5),… Joining them by means of a smooth

curve and extending it upwards we get the required graph. We notice that:

i) The entire graph lies above the x-axis.

ii) Two equal and opposite values of x corresponding to every value of y (but not vice versa).

iii) 0 As x increases (numerically) y increases and there is no end to their increase. Thus the graph goes infinitely upwards.

Rational Function

A rational function is a function that can be represented as the ratio of two polynomial functions.

Example: f(x) = (3x^2 + 2x – 1) / (x – 2) is a rational function.

Algebraic Functions

An algebraic function is a function that can be constructed using algebraic operations (addition, subtraction, multiplication, division) and applying roots and powers to the variable(s).

Example: f(x) = √(x^3) + 2x is an algebraic function.

Cubic Function

A cubic function is a polynomial function of degree 3. Its graph often has an S-shape.

Example: f(x) = x^3 – 4x^2 + 2x + 1 is a cubic function.

Even and Odd Function

An even function is a function that satisfies f(x) = f(-x) for all x in its domain.

 An odd function satisfies f(x) = -f(-x) for all x in its domain.

Example of an even function: f(x) = x^2

Example of an odd function: f(x) = x^3

Periodic Function

A periodic function is a function that repeats its values at regular intervals. There exists a positive constant ‘P’ such that f(x + P) = f(x) for all x in the domain.

Example: f(x) = sin(x) is a periodic function with a period of 2π.

Composite Function

A composite function is formed by applying one function to the output of another function.

Example: If f(x) = 2x and g(x) = x + 3, then the composite function (g ∘ f)(x) = g(f(x)) = g(2x) = 2x + 3.

Set-Builder Notation for a function

 We know that sub-builder notation is more

suitable for infinite sets. So is the case in respect of a function comprising an infinite

number of ordered pairs. Consider, for instance, the function

f = {(1,1), (2, 4), (3, 9), (4, 16)….}

Dom f= {1, 2, 3, 4,…}.

Ran f = {1,4,9, 16, …}

This function may be written as:

f = {(x, y) l y =r², x ∈ N}

For the sake of brevity this function may be written as:

f = function defined by the equation y=x2, x ∈ N.

Or, to be still more brief: The function, x2, x ∈ N.

In algebra and Calculus the domain of most functions is R and if evident from the context it is, generally, omitted.

 The inverse of a function

If a relation or a function is given in the tabular form i.e., as a set of ordered pairs, its inverse is obtained by interchanging the components of each ordered pair.

The inverse of r and f is denoted r-1 and f-1 respectively.

If r or f are given in set-builder notation the inverse of each is obtained by interchanging x and y in the defining equation. The inverse of a function may or may not be a function.

The inverse of the linear function

{(x, y) y= mx+ c) is ((x, y) l x= my+ c) which is also a linear function.

Briefly, we may say that the inverse of a line is a line.

The line y= x is clearly self-inverse. The function defined by this equation i.e., the function (x, y) | y= x) is called the identity function.


 Find the inverse of

i) {(1, 1), (2, 4). (3, 9), (4, 16), x ∈ Z+}

ii) {(x, y) l y= 2x+ 3, x ∈ R)

Which of these are functions.


i) The inverse is:

{(2, 1), (4, 2), (9,3), (16, 4)…}

This is also a function.

Note: Remember that the equation

y= √x, x ≥ 0

defines a function but the equation y2 = x, x ≥ 0 does not define a function.

The function defined by the equation.

y= √x, x ≥ 0

 is called the square root function.

The equation y2 = x ⇒ y = ±√x

Therefore, the equation y= x (x ≥ 0) may be regarded as defining the union of the functions defined by

y = √x, x ≥ 0 and y = -√x, x ≥ 0.

ii) The given function is a linear function.

Its inverse is: {(x, y) | x= 2y + 3}

Which is also a linear function.

Points (0, 3), (-1.5, 0) lie on the given line and points (3, 0), (0, -1.5) lie on its inverse.