Functions are relations where each input has a particular output. This lesson covers the concepts of functions in mathematics and the different types of functions using various examples for better understanding.
Contents Related to Functions
- Functions
- Limits, Continuity and Differentiability
- Differentiation
- Applications of Derivatives
JEE Main 2021 Maths LIVE Paper Solutions 24-Feb Shift-1 Memory-Based
What are Functions in Mathematics?
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets; mapping from A to B will be a function only when every element in set A has one end, only one image in set B.
Example:
Another definition of functions is that it is a relation “f” in which each element of set “A” is mapped with only one element belonging to set “B”. Also in a function, there can’t be two pairs with the same first element.
A Condition for a Function:
Set A and Set B should be non-empty.
In a function, a particular input is given to get a particular output. So, A function f: A->B denotes that f is a function from A to B, where A is a domain and B is a co-domain.
- For an element, a, which belongs to A, a ∈ A, a unique element b, b ∈ B is there such that (a,b) ∈ f.
The unique element b to which f relates a, is denoted by f(a) and is called f of a, or the value of f at a, or the image of a under f.
- The range of f (image of a under f)
- It is the set of all values of f(x) taken together.
- Range of f = { y ∈ Y | y = f (x), for some x in X}
A real-valued function has either P or any one of its subsets as its range. Further, if its domain is also either P or a subset of P, it is called a real function.
Vertical Line Test:
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 points then the curve is not a function.
Representation of Functions
Functions are generally represented as 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.
Steps for Solving Functions
Question: Find the output of the function g(t) = 6t2 + 5 at
(i) t = 0
(ii) t = 2
Solution:
The given function is g(t) = 6t2 + 5
(i) At t = 0, g(0) = 6(0)2 + 5 = 5
(ii) At t = 2, g(2) = 6(2)2 + 5 = 29
Types of Functions
There are various types of functions in mathematics which are explained below in detail. The different function types covered here are:
- One – one function (Injective function)
- Many – one function
- Onto – function (Surjective Function)
- Into – function
- Polynomial function
- Linear Function
- Identical Function
- Quadratic Function
- Rational Function
- Algebraic Functions
- Cubic Function
- Modulus Function
- Signum Function
- Greatest Integer Function
- Fractional Part Function
- Even and Odd Function
- Periodic Function
- Composite Function
- Constant Function
- Identity Function
Practice: Find the missing equations from the above graphs.
Functions – Video Lessons
Functions and Types of Functions
Number of Functions
Even and Odd Functions
Composite and Periodic Functions
One – one function (Injective function)
If each element in the domain of a function has a distinct image in the co-domain, the function is said to be one – one function.
For examples f; R R given by f(x) = 3x + 5 is one – one.
Many – one function
On the other hand, if there are at least two elements in the domain whose images are same, the function is known as many to one.
For example f : R R given by f(x) = x2 + 1 is many one.
Onto – function (Surjective Function)
A function is called an onto function if each element in the co-domain has at least one pre – image in the domain.
Into – function
If there exists at least one element in the co-domain which is not an image of any element in the domain then the function will be Into function.
(Q) Let A = {x : 1 < x < 1} = B be a mapping f : A B, find the nature of the given function (P). F(x) = |x|
Solution for x = 1 & -1
Hence, it is many one the Range of f(x) from [-1, 1] is [0, 1], which is not equal to the co-domain.
Hence, it is into function.
Lets say we have function,
\(\begin{array}{l}f(x)=\left\{\begin{matrix} x^2 & ; & x\geq 0\\ -x^2 & ; & x<0 \end{matrix}\right.\end{array} \)
For different values of Input, we have different output hence it is one – one function also it manage is equal to its co-domain hence it is onto also.
Polynomial function
A real-valued function f : P → P defined by
\(\begin{array}{l}y = f(a) = h_{0}+h_{1}a+…..+h_{n}a^{n}\end{array} \)
, where n ∈ N and h0 + h1 + … + hn ∈ P, for each a ∈ P, is called polynomial function.- N = a non-negative integer.
- The degree of the Polynomial function is the highest power in the expression.
- If the degree is zero, it’s called a constant function.
- If the degree is one, it’s called a linear function. Example: b = a+1.
- Graph type: Always a straight line.
So, a polynomial function can be expressed as :
\(\begin{array}{l}f(x)= a_{n}x^{n}+a_{n-1}x^{n-1}+…..+a_{1}x^{1}+a_{0}\end{array} \)
The highest power in the expression is known as the degree of the polynomial function. The different types of polynomial functions based on the degree are:
- The polynomial function is called a Constant function if the degree is zero.
- The polynomial function is called a Linear if the degree is one.
- The polynomial function is Quadratic if the degree is two.
- The polynomial function is Cubic if the degree is three.
Linear Function
All functions in the form of ax + b where a, b ∈ R & a ≠ 0 are called linear functions. The graph will be a straight line. In other words, a linear polynomial function is a first-degree polynomial where the input needs to be multiplied by m and added to c. It can be expressed by f(x) = mx + c.
For example, f(x) = 2x + 1 at x = 1
f(1) = 2.1 + 1 = 3
f(1) = 3
Another example of linear function is y = x + 3
Identical Function
Two functions f and g are said to be identical if
(a) The domain of f = domain of g
(b) The range of f = the Range of g
(c) f(x) = g(x) ∀ x ∈ Df & Dg
For example f(x) = x
\(\begin{array}{l}g(x) = \frac{1}{1/x}\end{array} \)
Solution: f(x) = x is defined for all x
But
\(\begin{array}{l}g(x) = \frac{1}{1/x}\end{array} \)
is not defined of x = 0Hence it is identical for x ∈ R – {0}
Quadratic Function
All functions in the form of y = ax2 + bx + c where a, b, c ∈ R, a ≠ 0 will be known as Quadratic function. The graph will be parabolic.
\(\begin{array}{l}\text{At}\ x=\frac{-b \pm \sqrt{D}}{2}\end{array} \)
, we will get its maximum on minimum value depends on the leading coefficient and that value will be -D/4a (where D = Discriminant)In simpler terms,
A Quadratic polynomial function is a second degree polynomial, and it can be expressed as;
F(x) = ax2 + bx + c, and a is not equal to zero.
Where a, b, c are constant, and x is a variable.
Example, f(x) = 2x2 + x – 1 at x = 2
If x = 2, f(2) = 2.22 + 2 – 1 = 9
For Example: y = x2
Read More: Quadratic Function Formula
Rational Function
These are the real functions of the type
\(\begin{array}{l}\frac{f(a)}{g(a)}\end{array} \)
where f (a) and g (a) are polynomial functions of a defined in a domain, where g(a) ≠ 0.- For example f : P – {– 6} → P defined by
\(\begin{array}{l}f(a) = \frac{f(a+1)}{g(a+2)}\forall a\in P – {-6},\end{array} \)
is a rational function. - Graph type: Asymptotes (the curves touching the axes lines).
Algebraic Functions
An algebraic equation is known as a function that consists of a finite number of terms involving powers and roots of independent variable x and fundamental operations such as addition, subtraction, multiplication, and division.
For Example,
\(\begin{array}{l}f(x)=5x^{3}-2x^{2}+3x+6\end{array} \)
,\(\begin{array}{l}g(x)=\frac{\sqrt{3x+4}}{(x-1)^{2}}\end{array} \)
.Cubic Function
A cubic polynomial function is a polynomial of degree three and can be expressed as;
F(x) = ax3 + bx2 + cx + d and a is not equal to zero.
In other words, any function in the form of f(x) = ax3 + bx2 + cx + d, where a, b, c, d ∈ R & a ≠ 0
For example: y = x3
Domain ∈ R
Range ∈ R
Modulus Function
The real function f : P → P defined by f (a) = |a| = a when a ≥ 0. and f(a) = -a when a < 0 ∀ a ∈ P is called the modulus function.
- Domain of f = P
- Range of f = P+ U {0}
\(\begin{array}{l}y=|x|=\left\{\begin{matrix} x & x\geq 0\\ -x & x<0 \end{matrix}\right.\end{array} \)
Domain: R
Range: [0, ∞)
Signum Function
The real function f : P → P is defined by
\(\begin{array}{l}\left\{\begin{matrix}\frac{\left | f(a) \right |}{f(a)}, a\neq 0 \\ 0, a=0 \end{matrix}\right. = \left\{\begin{matrix} 1,if x>0\\ 0, if x=0\\ -1, if x<0\end{matrix}\right.\end{array} \)
is called the signum function or sign function. (gives the sign of real number)
- Domain of f = P
- Range of f = {1, 0, – 1}
For example: signum (100) = 1
signum (log 1) = 0
signum (x21) =1
Greatest Integer Function
The real function f : P → P defined by f (a) = [a], a ∈ P assumes the value of the greatest integer less than or equal to a, is called the greatest integer function.
- Thus f (a) = [a] = – 1 for – 1 ⩽ a < 0
- f (a) = [a] = 0 for 0 ⩽ a < 1
- [a] = 1 for 1 ⩽ a < 2
- [a] = 2 for 2 ⩽ a < 3 and so on…
The greatest integer function always gives integral output. The Greatest integral value that has been taken by the input will be the output.
For example: [4.5] = 4
[6.99] = 6[1.2] = 2Domain ∈ R
Range ∈ Integers
Fractional Part Function
{x} = x – [x]
It always gives fractional value as output.
For example:- {4.5} = 4.5 – [4.5]
= 4.5 – 4 = 0.5
{6.99} = 6.99 – [6.99]
= 6.99 – 6 = 0.99
{7} = 7 – [7] = 7 –7 = 0
Even and Odd Function
If f(x) = f(-x) then the function will be even function & f(x) = -f(-x) then the function will be odd function
Example 1:
f(x) = x2sinx
f(-x) = -x2sinx
Here, f(x) = -f(-x)
It is an odd function.
Example 2:
\(\begin{array}{l}f(x)={{x}^{2}}\end{array} \)
and
\(\begin{array}{l}f(-x)={{x}^{2}}\end{array} \)
f(x) = f(-x)
It is an even function.
Periodic Function
A function is said to be a periodic function if a positive real number T exists, such that f(u – t) = f(x) for all x ε Domain.
For example f(x) = sin x
f(x + 2π) = sin (x + 2π) = sin x fundamental
then period of sin x is 2π
Composite Function
Let A, B, C be three non-empty sets
Let f: A → B & g : G → C be two functions, then gof : A → C. This function is called the composition of f and g given gof (x) = g(f(x)).
For example f(x) = x2 & g(x) = 2x
f(g(x)) = f(2x) = (2x)2 = 4x2
g(f(x)) = g(x2) = 2x2
Constant Function
The function f : P → P is defined by b = f (x) = D, a ∈ P, where D is a constant ∈ P, is a constant function.
- Domain of f = P
- Range of f = {D}
- Graph type: A straight line which is parallel to the x-axis.
In simple words, the polynomial of 0th degree where f(x) = f(0) = a0 = c. Regardless of the input, the output always results in a constant value. The graph for this is a horizontal line.
Identity Function
P= set of real numbers
The function f : P → P defined by b = f (a) = a for each a ∈ P is called the identity function.
- Domain of f = P
- Range of f = P
- Graph type: A straight line passing through the origin.
Functions Video
Domain, Range, Period of Functions
Functions and Relations
Relations and Functions Questions
One-One and Onto Functions
Frequently Asked Questions
A relation f from set A to set B is called a function if every element of set A has one and only one image in set B. The domain of a function is the set of all possible inputs for a function. The range of a function is the set of all possible output values. Constant function is a function whose output is the same for every input value. For example, f(x) = 3. Here for every value of x, output will be 3.What do you mean by a function in Mathematics?
What do you mean by domain of a function?
What do you mean by range of a function?
What do you mean by a constant function?
What are the 4 types of functions?
What are the 7 types of functions?
What are the 12 types of functions?
What are the 2 types of functions give examples?