uksin

ghmaths.com

uksinghmaths.com

Functions and Relations :

uksin

ghmaths.com

uksinghmaths.com

Some basic concepts for understanding functions :

Variables :- Variables are those whose values does not remain same during any mathematical discussions. Variables are generally denoted by x,y,z etc. (Generally last letters of English alphabet).

Constant :- Constants are those whose values always remains the same during any mathematical discussions. Constants are denoted by a,b,c,k etc. (Generally initial letters of English alphabet) .

Preliminary definition of Function :

If x and y are two variables so related that for each value of x there is a definite value of y then y is called a function of x and we write y = f(x) or y = y(x).

for example, y =x2 when, x = 0, y = 0 x = ±1, y = 1 x = ±2, y = 4 ………………… ………………… x is called independent variable and y is called dependent variable.

If for some values of x the value of y is not determinate then we say that the function is not defined for those values of x.

uksin

ghmaths.com

uksinghmaths.com

For example, y = 1/x is not defined for x = 0, because when x = 0 then y = ∞ (infinity) , which is not definite.

Another Example, Let y= tanx. This function is not defined for x = π/2.

The totality of the values of x is called Domain of the function .

The totality of the values of y is called Range of the function.

Function is one of the most important concepts in mathematics. Function can be visualised as a rule which produces new elements out of some given elements. It is also known as mapping.

The word ‘function’ is derived from a Latin word meaning Operation. Thus when we square a given real number x, we think of performing an operation on real number x, to obtain a positive real number x2 .

The concept o f func t ion i s exp ressed as a correspondence or association between two sets . It can also be expressed as a special type of relation from one set to another set. Two sets under consideration may be same set or may be distinct sets.

uksin

ghmaths.com

uksinghmaths.com

Let A and B are two non-empty sets. Let there be a correspondence between the elements of A and B by any rule whatsoever such that : (i) Every elements of A corresponds to ( or associated to)

some element of B . (ii) Each element of A corresponds to (or associated to)

Unique element of B. Means, No element of A corresponds to more than one elements of B.

(iii)Such a correspondence which associate the elements of set A to set B is called function or mapping from A to B and is denoted by f and written as f : A➝B . f : A➝B is read as “ f is such that A is mapped upon B ” .

Another definition of function in terms of Relation: Let A and B be two non-empty sets. A function f is a relation from set A to set B such that domain of f is A and no two ordered pairs in f have the same first component. Means, A function f from a set A into a set B is a relation from A to B if for each a∊A there exists unique b∊B such that (a,b) ∊ f .

The graph of a function f : A→B is the set of points (a,f(a)) in A×B, where a∈A.

Note :- f is a function NOT f(a) . f(a) is the value of function f at a .

FUNCTION (or MAPPING) :

uksin

ghmaths.com

uksinghmaths.com

Let us consider two sets given below A = { a,b,c,d } and B = { x,y,z,u,v }

let f be a rule which associates elements of A and B in the following manner, f(a) = x, f(b) = y, f(c) = u and f(d) = x ; Then f is a function or mapping from A to B . Let us represent this function diagrammatically as follows :

!

Remarks :

More than one elements of A may corresponds to one element of B.

There may be some elements in B which may not be corresponded by any elements to A.

a

b

c

d

x

y

z

u

v

fA B

uksin

ghmaths.com

uksinghmaths.com

Let us us consider two sets A and B consisting of some mythological characters (shown in diagram) from epic Ramayana and let f be a rule that associates the elements of A to B and defined in the following ways

f : A➝B by f(x) = { y∊B : y is husband of x, where x∊A }

!

Then, clearly this rule f is a function from A to B. As more than one elements of A may corresponds to one element of B .There may be some elements in B which may not be corresponded by any elements to A.

Kausalya

Sumitra

Kaikeyi

Sita

Urmila

Dashratha

Ram

Lakshmana

Bharata

Shatrughan

fA B

uksin

ghmaths.com

uksinghmaths.com

Now let us consider two sets X and Y consisting of some mythological characters (shown in diagram) from another epic Mahabharata and let g be a rule that associates the elements of X to Y and defined in the following ways :

g : X➝Y by g(u) = { v∊Y : v is husband of u, where u∊x }

!

Then, Clearly this rule g is not a function from X to Y. As two necessary conditions for being a function is dishonoured over here.

Draupadi

Uttara

Gandhari

Rukmini

Yudhishtira

Bhima

Arjuna

Nakula

Sahadeva

Dhritarashtra

Abhimanyu

Bhishma

gX Y

uksin

ghmaths.com

uksinghmaths.com

Firstly, No element of X corresponds to more than one elements of Y, which is violated by an element of X namely Draupadi.

Secondly, Every elements of A must corresponds to some element of B, which is violated by an element of X namely Rukmini.

Image and pre-image under a function : If under a function f the element a∊A, corresponds to an element x∊B . Then ‘x is called image of a’ and we write it as x = f(a).

Also, ‘a is called pre-image of x’ .

Domain, Co-domain & Range of a Function: Let f : A➝B be a function (mapping) from A to B. ✦ Then, the set A is called domain of the function or

mapping f .

✦ The set B is called co-domain of the function or mapping f.

✦ The set of all images of the elements of A under function f is called Range of the function or mapping f & is denoted by f(A) and is given by

Range of f = f(A) = { f(x) : x∊A }

Therefore, mathematically to define a function, one has to provide its domain, co-domain and the images of elements in its domain either by a general rule (formula) or by listing then one by one.

uksin

ghmaths.com

uksinghmaths.com

Q. Let N be the set of natural numbers and f be a rule associating the elements of N to elements of N and defined as f(x) = 3x-5 . Check whether this rule is a function or not.

Soln : we have, the set N is given by N = { 1,2,3,4,5,6,…………………} If f is a function from N to N, then every elements of N must have a unique image in N. let us find the image of 1∊N. then, according to rule f, the image of 1 is given by : f(1) = 3.1-5 = -2 ∉ N Thus, the rule f is not a function from N to N .

Q. Let A={1,2,3,4,5} and N={1,2,3,4,5,…………..} be the two sets and f : A➝N be a mapping from A to N defined by f(x) = 2x+1 .

(i) find the Range of f, (ii) pre-image of 5 and 9 .

Soln : we have, the function defined by f(x) = 2x+1 Hence, f(1) = 2.1+1 = 3 f(2) = 2.2+1 = 5 f(3) = 2.3+1 = 7 f(4) = 2.4+1 = 9 f(5) = 2.5+1 = 11

So, the Range of the function f is given by

f(A) = { f(x) : x∊A } = {3,5,7,9,11}

uksin

ghmaths.com

uksinghmaths.com

Clearly, the pre-image of 5 is 2 and pre-image of 9 is 4 . This can be evaluated as follows : let x be the pre-image of 5. Then according to definition f(x) = 5 ⟹ 2x+1 = 5 ⟹ 2x = 5-1 = 4 ⟹ x = 4/2 = 2 . lly, let y be the pre-image of 9 then f(y) = 9 ⟹ 2y+1 = 9 ⟹ 2y = 9-1 = 8 ⟹ y = 8/2 = 4 .

Equal Functions :

Let f and g be two functions then, f is said to be equal to g if and only if it satisfies following three conditions : (i) domain of f = domain of g (ii) co-domain of f = co-domain of g (iii) images of every elements of their common domain are

same, i.e. f(x) = g(x) , ⩝ x ∊ common domain .

Real valued functions of a real variable (Real functions) :

Real valued function : A function f : X➝Y is called real valued function, if B is a subset of R ( set of all real numbers ) .

Real valued function of a real variable : Functions whose domain and co-domain are both subsets of R ( the set of all real numbers ), are called real valued functions of a real variable, or simply real functions.

uksin

ghmaths.com

uksinghmaths.com

Types of functions : Following are some of the frequently used standard real valued functions :

uksin

ghmaths.com

uksinghmaths.com

Let us elaborate some very important functions which are used frequently.

Into function/mapping : A mapping f : A➝B, is called an into function if there exists at least one element in B which is not the f-image of any element of A. for example, Let A = { 1,2,3 } and B= { 4,5,6,7 } let f : A➝B be defined by f(x) = x+3

! This mapping is an into mapping as 7 is not the f-image of any element of A.

Another example, Let I be the set of all integers and N- be the set of all non-negative integers . Let f : I➝ N- be defined by f(x) = x2,Then this mapping is an into mapping.

1

2

3

4

5

6

7

fA B

uksin

ghmaths.com

uksinghmaths.com

Onto function or Surjective function : The function f : A➝B, is called an Onto function/Surjective function if every element in B is f-image of some element of A. Thus for onto function the range of f is co-domain B, i.e. f(A) = B . for example, Let N be the set of all natural numbers and A be the set {1,-1} . Let f : N➝A be defined by f(x) = 1, if x is even. = -1, if x is odd.

!

This function is onto as every element of A is image of some element of N .

An another example, Let R be set of all real numbers and R+ , be the set of all positive real numbers. Let f : R➝ R+ , be defined by f(x) = ex . we check whether a function is onto or not in the following ways :

12345. .

1

-1

fN A

uksin

ghmaths.com

uksinghmaths.com

Let y ∊R+ be arbitrary .

⟹ logy is a real number ⟹ logy ∊ R Now, f(logy) = elogy = y . Hence, y is the image of logy ∊ R . ⟹ The function is onto function .

Many one function/mapping : Let f : A➝ B, be a function from A to B, if more than one element of A corresponds to same element of B then the function is called many one function . for example, Let X be set of all integers and Y be the set of non-negative integers . Let f : X➝ Y, be defined by f(x) = x2 , then the function is many one .

!

01-12-2. . .

0 1 2 3 4 5 . .

fX Y

uksin

ghmaths.com

uksinghmaths.com

Injective function or One-one function :

Let f : A➝B be a function from A to B. If different elements of A have different images in B then the function is called one-one function.

for example, Let A = { 1,2,3,4 } B= { 1,4,9,16 } Let f : A➝B be defined by f(x) = x2 .

Then the function is one-one.

!

1

2

3

4

1

4

9

16

fA B

uksin

ghmaths.com

uksinghmaths.com

Let us consider another example, Let R+ be the set of all positive real numbers and R be the set of all real numbers and consider the function f : R+ ➝ R , defined by f(x) = log x . Let us check whether the function f is one-one or not . f(x) = f(y) ⟹ log x = log y ⟹ x = y ⟹ The function is one-one .

Note:- In order to show that the function is one-one . we must show that either x ≠ y ⟹ f(x) ≠ f(y) or f(x) = f(y) ⟹ x = y .

Bijective function or One-one onto function or One-one correspondence function :

Let f : A ➝ B be a function from A to B . If the function f is both one-one and onto then, this is called a bijective function or one-one correspondence .

In case of finite sets, the function will be bijective if both sets have same finite numbers of elements .

for example,

Let A be the set of all Natural numbers and B be the set of all natural squares . Then, f : A ➝ B be defined by f(x) = x2 , is a one-one onto function.

uksin

ghmaths.com

uksinghmaths.com

Let us consider an another example, Let R be the set of all real numbers and R+ be the set of all positive real numbers . Let f : R ➝ R+ , be defined by f(x) = ex .

Test for one-one : f(x) = f(y) ⟹ ex = ey ⟹ x = y ⟹ The function is one-one.

Test for Onto : Let x∊R+ ⟹ x is a positive real number .

1

2

3

4. .

1

4

9

16 . .

fA B

uksin

ghmaths.com

uksinghmaths.com

⟹ logx is a real number .⟹ logx ∊ R .

Now , f(logx) = elogx = x . ⟹ x ∊ R+ is image of logx ∊ R . ⟹ The function f is onto.Since the function f is both one-one and onto, so it is a bijective function.

Inverse function/mapping :

Let f : A ➝ B be a function from A to B, such that i) f is one-one . ii) f is onto . Then we define a function from B to A , the function is denoted by f-1. The function is defined as follows : f-1(y) = x where y = f(x). This function f-1 is called inverse function of f.

���

x f(x) = y

f

A B

f-1

uksin

ghmaths.com

uksinghmaths.com

To define an inverse function of a given function, the given function must be one-one and onto. for Let f : A ➝ B be a function from A to B .

If f is not one-one then at least one element of B will have more than one image in A under function f-1 and therefore f-1 will not be a function .

If f is not onto then there will be at least one element in B which will have no image in A under the function f-1

and so f-1 will not be a function .

x1

x2

y1

A Bf-1

x1x2x3

y1 y2 y3 y4 y5

A Bf-1

uksin

ghmaths.com

uksinghmaths.com

Explicit function : If in a function the independent variable and dependent variable are easily separable then the function is called explicit function.

for example, y=x2

x+y-6=0 x2+y2=a2 are explicit function.

Implicit Function: A function is said to be implicit function if the dependent variable and independent variable are not easily separable.

for example, ax2+2h.x.y+hy2=1 (x+y)m+n =xm. yn sin(x+y)=tan(x.y) are implicit functions.

Parametric Function : If the dependent variable and independent variables both are functions of a third variable then the function is called parametric functions and third variable is called parameter .

for example, x=a cosθ, y=a sinθ and x=at2, y=2at ; x=a cosθ, y=b sinθ etc. are parametric functions.

uksin

ghmaths.com

uksinghmaths.com

Now, x= a cosθ, y= a sinθ ⇒ x2+y2= a2 cos2θ+a2 sin2θ

⇒ x2+y2= a2

⇒ x= a cosθ, y= a sinθ; is the parametric

representation of equation of circle, whose centre is origin(0,0) and radius is ‘a’ units. Here, θ is called parameter.

Similarly, x= at2, y= 2at ⇒ y2= 4a2t2

⇒ y2= 4a(at2)

⇒ y2= 4ax

⇒ x= at2, y= 2at; is the parametric representation

of equation of right handed parabola whose verses is origin(0,0) and focus is (a,0). Here, t is called parameter.

lly, x= a cosθ, y= b sinθ ⇒ x/a= cosθ, y/b= sinθ

⇒ x2/a2+y2/a2= 1 ⇒ x= a cosθ, y= b sinθ; is the parametric

representation of equation of ellipse whose centre is origin(0,0), vertices are (± a,0) and foci are (± ae,0), where e is the eccentricity of ellipse. Here, θ is called parameter.

uksin

ghmaths.com

uksinghmaths.com

Even Function : A function f(x) of x is said to be an even function of x if f(-x) = f(x) . for example, Let us consider a function f(x)= x2 . f(x)= x2 is an even function, for f(-x)= (-x)2 =x2 = f(x) .

Let us consider function f(x)= cosx . f(x)= cosx, is also an even function, for f(-x)= cos(-x)=cosx= f(x) .

Let us consider function f(x)= x.sinx . f(x)= x.sinx, is an even function, for f(-x)= -x.sin(-x) = x.sinx= f(x) .

Odd Function : A function f(x) of x is said to be odd function of x if f(-x) = - f(x) . for example, Let us consider a function f(x)= x3 . f(x) = x3, is an odd function, for f(-x)= (-x)3= -x3= -f(x) .

Let us consider function f(x)= sinx . f(x)= sinx, is also an odd function, for f(-x)= sin(-x)= - sinx= - f(x) .

Let us consider another function f(x)= x.cosx . f(x)=x.cosx, is an odd function, for f(-x)= -x.cos(-x) = -x. cosx = f(x) .

uksin

ghmaths.com

uksinghmaths.com

Periodic Function : A function f(x) of x is said to be periodic function if f(a+x) = f(x) . Where ‘a’ is called period of the function. for example, Let us consider the function f(x)= sinx . The function f(x)= sinx, is a periodic function, as f(2π + x) = sin(2π + x) = sinx = f(x) . 2π is the period of the function.

Q. If f(x)=1+x / 1-x show that f(tanx) = tan(π/4+x) Soln : As given f(x) = 1+x/1-x

f(tanx) = (1+tanx) /(1-tanx) = (tanπ/4 + tanx) / (1- tanπ/4. tanx) = tan(π/4+x)

uksin

ghmaths.com

uksinghmaths.com

Composition of functions : Let A, B, C be three sets . Let f : A ➝ B and g : B ➝ C be two functions . We define a function gof : A ➝ C by gof (x) = g(f(x)) , ⩝ x∊A . This function gof (x) is called composition of two functions f and g. In general composition of two functions are not commutative. i.e, gof ≠ fog

This is also called as product of two functions f and g . we can write gof simply as gf and fog simply as fg .

�

f g

gof

xf(x)

A c

g(f(x))

B

uksin

ghmaths.com

uksinghmaths.com

Q. Given that f(x) = x2 and g(x) = sinx . Prove that gof ≠ fog .

Now, gof(x) = g(f(x)) = g[f(x)]

= g[x2] = sin(x2)………..(i)

and, fog(x) = f(g(x)) = f[g(x)] = f[sinx] = (sinx)2

= sin2x ………….(ii) from (i) and (ii), it is clear that gof ≠ fog