Types of Functions JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Types of Functions JEE Notes | EduRev

The document Types of Functions JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
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FUNCTIONS
We can define a function as a special relation which maps each element of set A with one and only one element of set B. Both the sets A and B must be non -empty. A function defines a particular output for a particular input. Hence, f: A → B is a function such that for a ∈ A there is a unique element b ∈ B such that (a, b) ∈ f

  • The set ‘A’ is called the domain of ‘f’.
  • The set ‘B’ is called the co-domain of ‘f’.
  • Set of images (outputs) of different elements of the set A is called the range of ‘f’. It is obvious that range could be a subset of the co-domain as we may have some elements in the co-domain which are not the images of any element of A (of course, these elements of the co-domain will not be included in the range). Range is also called domain of variation.
  • If R is the set of real numbers and A and B are subsets of R, then the function f (A) is called a real−valued function or a real function.

Domain of a function ‘f’ is normally represented as Domain (f). Range is represented as Range (f). Note that some times domain of the function is not explicitly defined. In these cases domain would mean the set of values of ‘x’ for which f (x) assumes real values that is if y = f (x) then called Domain (f) = {x : f (x) is a real number}.

In other words, domain is defined as a set of all those values of x for which the given function is defined.

Q. Find the domain and range of the function 
Types of Functions JEE Notes | EduRev
Ans. The function
Types of Functions JEE Notes | EduRev
is defined for x ≥ 5
⇒ The domain is [5, ∞] .
Also, for any
Types of Functions JEE Notes | EduRev
⇒ The range of the function is [0, ∞] .

Q. Find the domain and the range of the function y = f (x), where f (x) is given by
(i) x2 − 2x − 3

Types of Functions JEE Notes | EduRev
Types of Functions JEE Notes | EduRev
(iv) tan x
(v) tan−1x
(vi) log10 (x).
(vii) sin x

Ans. 
(i) Here y = (x − 3) (x + 1).
The function is defined for all real values of x
⇒ Its domain is R.
Also x2 − 2x − 3 − y = 0 for real x
⇒ 4 + 4 (3 + y) ≥ 0 ⇒ − 4 ≤ y < ∞.
Hence the range of the given function is [- 4, ∞].
(ii) Here
Types of Functions JEE Notes | EduRev

⇒ (x − 3) (x + 1) ≥ 0 so that x ≥ 3 or x ≤ − 1.
Hence the domain is R − (− 1, 3) or (− ∞, - 1] ∪ [3, ∞].

Since f(x) is non−negative in the domain, the range of f (x) is the interval [0, ∞].
(iii) The given function is defined for all x ≥ 0
⇒ The domain is [0, ∞].
Moreover - 1 ≤ f (x) < ∞
⇒ The range is [- 1, ∞].
(iv) The function f (x) = tan x = sin x/cos x is not defined when
cos x = 0, or x = (2n + 1) π/2, n = 0, ± 1, ± 2, …….
Hence domain of tan x is R -{(2n + 1)π/2, n = 0, + 1, + 2,....}, and its range is R. 

(v) The function f (x) = tan−1 x, is defined for all real values of x and -π/2 < tan-1 < π/2.
Hence its domain is R and the range is (-π/2, π/2).
(vi) The function f (x) = log10 x is defined for all x > 0. Hence its domain is (0, ∞) and range is R. 

(vii) The function f (x) = sin x, (x in radians) is defined for all real values of x
⇒ domain of f (x) is R. Also − 1 ≤ sin x ≤ 1, for all x,
so that the range of f (x) is [− 1, 1].

TYPES OF FUNCTIONS
We have already learned about some types of functions like Identity, Polynomial, Rational, Modulus, Signum, Greatest Integer functions. In this section, we will learn about other types of function.
One to One Function
A function f: A → B is One to One if for each element of A there is a distinct element of B. It is also known as Injective. Consider if a1 ∈ A and a2 ∈ B, f is defined as f: A → B such that f (a1) = f (a2) f is defined as f: A → B such that f(x) = y then

Types of Functions JEE Notes | EduRev

f: A → B is said to be one-to one function if 
f (a1) = f (a2
⇒ (a1) = (a2), (a1)(a2) ∈ A 
Let A = {(a1)(a2)(a3)(a4)} and B = {(b1)(b2)(b3)(b4
)}

Types of Functions JEE Notes | EduRev

In other words if a1, a2 ∈ A such that a1 ≠ a2  then f(a1) ≠ f(a2)

Solved examples of one to one function
Q.1. Determine if the function given below is one to one. 
(i) To each state of India assign its Capital
Ans. This is not one to one function because each state of India has not different capital.

(ii) Function = {(2, 4), (3, 6), (-1, -7)} 
Ans. The above function is one to one because each value of range has different value of domain.

(iii) f(x) = |x|
Ans. Here to check whether the given function is one to one or not, we will consider some values of x (domain) and from the given function find the value of range(y).

Types of Functions JEE Notes | EduRev
From the above table we can see that an element in the range repeats, then this is not a 1 to 1 function.

Q.2. Without using graph prove that the function
F : R → R defiend by f (x) = 4 + 3x is one-to-one
Ans. For a function to be one-one function if
F (x1) = F (x1) ⇒ xx2  ∀  x1, x2  ∈ domain
 Now f (x1) = f (x2) gives
4 + 3x1 = 4 + 3x2 or x1 = x2 
∴ F is a one-one function.

Many to One Function
It is a function which maps two or more elements of A to the same element of set B. Two or more elements of A have the same image in B.

Types of Functions JEE Notes | EduRevQ.3. Without using graph check the following function f: R → R defined by f (x) = x2 is one-to-one or not ?
Ans. For the function to be one to one , different elements must have different images
But here if x = 1 then f(1) = (1)2 =1
And if x = -1 then f(-1) =(-1)2 = 1
Clearly, x = 1 and x = -1 both have same image
So given function is not one to one i.e many to one function
In other words, we can say that a function which is not one to one that can be known as many to one.

Onto Function
If there exists a function for which every element of set B there is (are) pre-image(s) in set A, it is Onto Function. Onto is also referred as Surjective Function.
i.e ∀ y ∈ B, ∃ at least one element x ∈ A such that f(x) = y

Types of Functions JEE Notes | EduRev

e.g., let A = {2, 3, 4}, B= {5, 6} . Then f : A → B defined by f = {(2, 5), (3, 6),
(4, 5)} is an onto function as each element of B is the image of at least one element of A.
Note: A function which is not onto is known as into function.


Q.4. If f: R → R is defined as f (x)= 3x + 7, x ∈ R, then show that f is an onto function.
Ans. We have f (x) = 3x + 7, x ∈ R . Let b ∈ R so that f (x) = b ⇒ 3x + 7 = b or
x = (b - 7)/3. Since b ∈ R, (b - 7)/3 ∈ R.
Types of Functions JEE Notes | EduRev
⇒ b is image of (b - 7)/3 where b is arbitrary.
Hence f (x) is an onto function.

Q.5. Let f: N → I be a function defined as f(x) = x – 1000. Determine whether f(x) is into or onto.
Ans.

Let y ∈ I such that f(x) = y ,where x∈ N
⇒ x – 1000 = y
⇒ x = y + 1000 = g(y)
Here g(y) is defined for each y ∈ I, but g(y) ∉ N for y ≤ – 1000.
Hence f(x) is into.

One – One and Onto Function
A function, f is One – One and Onto or Bijective if the function f is both One to One and Onto function. In other words, the function f associates each element of A with a distinct element of B and every element of B has a pre-image in A.

Types of Functions JEE Notes | EduRev
Q. Prove that
F : R → R defined by f (x) = 4x3 - 5 is a bijection

Ans. Now f (x1) = f (2) ∀ x1, x2 Domain
∴ 4x3 - 5 = 4x23 - 5
⇒ x= x23
⇒ x- x2= 0 ⇒ (x- x1) (x12 + x1x2 + x22) = 0
⇒ x= xor
x12 + x1x2 + x2= 0 (rejected).
It has no real value of x1 and x2.
∴ F is a one-one function.
Again let y = (x) where y ∈ codomain, x ∈ domain.
We have y = 4x3 - 5
or  
Types of Functions JEE Notes | EduRev
∴ For each y ∈ codomain ∃ x ∈ domain such that f (x) = y.
Thus F is onto function.
∴ F is a bijection.

Other Types of Functions
A function is uniquely represented by its graph which is nothing but a set of all pairs of x and f(x) as coordinates. Let us get ready to know more about the types of functions and their graphs.

Q. Prove that F: R → R defined by F (x) = x2 + 3 is neither one-one nor onto function.
Ans. We have F (x1) = F (x2) ∀ x1, x2 domain giving
x+ 3 = x2+ 3 ⇒ x1= x22
or x1- x2= 0 ⇒ x= x2  or x= x2
or F is not one-one function.
Again let y = F (x) where y ∈ codomain
x ∈ domain.
⇒ y = x+ 3
Types of Functions JEE Notes | EduRev
⇒ ∀ y < 3 no real value of x in the domain.
∴ F is not an onto function.

Identity Function
Let R be the set of real numbers. If the function f: R → R is defined as f(x) = y = x, for x ∈ R, then the function is known as Identity function. The domain and the range being R. The graph is always a straight line and passes through the origin.

Types of Functions JEE Notes | EduRev
Solved Example of Identity Functions
Example:
Let A = {1, 2, 3, 4, 5, 6}
Then Identity function on set A will be defined as
IA : A → A , IA = x , x ∈ A
for x = 1 , IA(1) = x = 1
for x = 2 , IA(1) = x = 2
for x = 3 , IA(1) = x = 3
for x = 4 , IA(1) = x = 4
for x = 5 , IA(1) = x = 5
Domain, Range and co-domain will be Set A

Constant Function
If the function f: R→R is defined as f(x) = y = c, for x ∈ R and c is a constant in R, then such function is known as Constant function. The domain of the function f is R and its range is a constant, c. Plotting a graph, we find a straight line parallel to the x-axis.

Types of Functions JEE Notes | EduRev
Solved examples of constant functions
Q.1. which is below function is a constant function?
(a) y = x
(b) y = 11
(c) y = π
(d) x + y = 1
(e) y = x2
Ans.
For the function to be a constant function ,it should yield same value for each x
(a) This is constant function as output is different for each input
(b) This is constant function as output is same for each input. It is of the format y = c
(c) This is constant function as output is same for each input. It is of the format y = c
(d) This is constant function as output is different for each input
(e) This is constant function as output is different for each input

Q.2. which of the graph represent constant function?
Types of Functions JEE Notes | EduRevAns.
The graph should be parallel to x-axis for the function to be constant function
So C and D are constant function.

Polynomial Function
A polynomial function is defined by y = a+ a1x + a2x2 + … + anxn, where n is a non-negative integer and a0, a1, a2,…, n ∈ R. The highest power in the expression is the degree of the polynomial function. Polynomial functions are further classified based on their degrees:
Constant Function: If the degree is zero, the polynomial function is a constant function (explained above).

Linear Function: The polynomial function with degree one. Such as y = x + 1 or y = x or y = 2x – 5 etc. Taking into consideration, y = x – 6. The domain and the range are R. The graph is always a straight line.Types of Functions JEE Notes | EduRev


Quadratic Function: If the degree of the polynomial function is two, then it is a quadratic function. It is expressed as f(x) = ax2 + bx + c, where a ≠ 0 and a, b, c are constant & x is a variable. The domain and the range are R. The graphical representation of a quadratic function say, f(x) = x2 – 4 is

Types of Functions JEE Notes | EduRev

Cubic Function: A cubic polynomial function is a polynomial of degree three and can be denoted by f(x) = ax3 + bx2 + cx +d, where a ≠ 0 and a, b, c, and d are constant & x is a variable. Graph for f(x) = y = x3 – 5. The domain and the range are R.

Types of Functions JEE Notes | EduRev

Rational Function: A rational function is any function which can be represented by a rational fraction say, f(x)/g(x) in which numerator, f(x) and denominator, g(x) are polynomial functions of x, where g(x) ≠ 0. Let a function f: R → R is defined say, f(x) = 1/(x + 2.5). The domain and the range are R. The Graphical representation shows asymptotes, the curves which seem to touch the axes-lines.

Types of Functions JEE Notes | EduRev


Modulus Function
The absolute value of any number, c is represented in the form of |c|. If any function f: R→ R is defined by f(x) = |x|, it is known as Modulus Function. For each non-negative value of x, f(x) = x and for each negative value of x, f(x) = -x, i.e.,
f(x) = {x, if x ≥ 0; – x, if x < 0.
Its graph is given as, where the domain and the range are R.

Types of Functions JEE Notes | EduRev

Signum Function
A function f: R→ R defined by
f(x) = {1, if x > 0; 0, if x = 0; -1, if x < 0}
Signum or the sign function extracts the sign of the real number and is also known as step function.

Types of Functions JEE Notes | EduRevGreatest Integer Function
If a function f: R→ R is defined by f(x) = [x], x ∈ X. It round-off to the real number to the integer less than the number. Suppose, the given interval is in the form of (k, k + 1), the value of greatest integer function is k which is an integer. For example: [-21] = 21, [5.12] = 5. The graphical representation is

Types of Functions JEE Notes | EduRev

Solved Example
Q. Which of the following is a function?

Types of Functions JEE Notes | EduRev

Types of Functions JEE Notes | EduRev

Types of Functions JEE Notes | EduRev

Ans. Figure (iii) is an example of a function. Since the given function maps every element of A with that of B. In figure (ii), the given function maps one element of A with two elements of B (one to many). Figure (i) is a violation of the definition of the function. The given function does not map every element of A.

ALGEBRA OF REAL FUNCTIONS
Real-valued Mathematical Functions
In mathematics, a real-valued function is a function whose values are real numbers. It is a function that maps a real number to each member of its domain. Also, we can say that a real-valued function is a function whose outputs are real numbers i.e., f: R → R (R stands for Real).

Types of Functions JEE Notes | EduRev
Algebra of Real Functions
In this section, we will get to know about addition, subtraction, multiplication, and division of real mathematical functions with another.

Addition of Two Real Functions
Let f and g be two real valued functions such that f: X → R and g: X → R where X ⊂ R. The addition of these two functions (f + g) : X → R  is defined by:

(f + g) (x) = f(x) + g(x), for all x ∈ X

Subtraction of One Real Function from the Other
Let f : X → R and g : X → R be two real functions where X ⊂ R. The subtraction of these two functions (f – g): X → R  is defined by:

(f – g) (x) = f(x) – g(x), for all x ∈ X

Multiplication by a Scalar
Let f: X → R be a real-valued function and γ be any scalar (real number). Then the product of a real function by a scalar γf: X → R is given by:
(γf) (x) = γ f(x), for all x ∈ X.

Equal functions 

Let f and g be two functions defined from A to B. Then f , g : A → B are equal if f (x) = g(x), x ∈ A .
If the function f and g are equal, then the subsets, graph of f and graph of g, of A x B are equal.

Q.1. 
Types of Functions JEE Notes | EduRev
find the value of k
Ans. We need to consider only one equation
2k = 6
k = 3

Q.2. Find the values of x and y.
Types of Functions JEE Notes | EduRev
Ans:
2x – 6 = 5
2x = 11
x = 5.5
4 – y = 3
y = 1

Multiplication of Two Real Functions
The product of two real functions say, f and g such that f: X → R and g: X → R, is given by

(fg) (x) = f(x) g(x), for all x ∈ X

Division of Two Real Functions
Let f and g be two real-valued functions such that f: X → R and g: X → R where X ⊂ R. The quotient of these two functions (f  ⁄ g): X → R  is defined by:

(f / g) (x) = f(x) / g(x), for all x ∈ X, but g(x) ≠ 0 for all x ∈ X

Note: It is also called point wise multiplication.

Solved Example
Q. Let f(x) = x3 and g(x) = 3x + 1 and a scalar, γ= 6. Find
(a) (f + g) (x)
(b) (f – g) (x)
(c) (γf) (x)
(d) (γg) (x)
(e) (fg) (x)
(f) (f / g) (x)

Sol: We have,
(a) (f + g) (x) = f(x) + g(x) = x+ 3x + 1.
(b) (f – g) (x) = f(x) – g(x) = x– (3x + 1) = x– 3x – 1.
(c) (γf) (x) = γ f(x) = 6x
(d) (γg) (x) = γ g(x) = 6 (3x + 1) = 18x + 6.
(e) (fg) (x) = f(x) g(x) = x(3x +1) = 3x4 + x3.
(f) (f / g) (x) = f(x) / g(x) = x/ (3x + 1), provided x ≠ – 1/3.

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