Function Lecture - Mathematics Notes, Study notes of Mathematics

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Function

1. Function

Definition of Set:

A set is a collection of distinct objects, considered

as an object in its own right.

Ex.

A = {1, 2, 3, 4}; B = {a, e, i, o, u}.

Note:

1. A set is generally represented by a capital

letter.

2. The elements of set are written within the

braces.

3. The numbers 2, 4 and 6 are distinct objects

when considered separately, but when they

are considered collectively, they form a single

set of size three, written {2, 4, 6}.

4. No element in the set is repeated. 5. Set is a collection in which order of elements

is not important.

Ex. {1, 2, 3 } ≡{3, 2, 1 .}

Function

Roster form

Representation of a set that lists all the elements in

the set, separated by commas, within braces.

Ex. {–3, –2, –1, 0, 1, 2}

Set Builder form

Mathematical notation for describing a set by

enumerating its elements or stating the properties

that its members must satisfy.

Ex. {x | – 4 ≤ x < 3, x ∈ R}

Know the facts

There is no set like {1, 2, 1}.

3. Function

Note:

It can be seen that all elements of

A × B and B × A are not equal.

∴ (^) A × B ≠ B × A

A brief introduction to ‘‘Relation’’

Any subset of A × B is a ‘‘Relation’’, from A →B

(pronounced A to B).

Ex.

If n (A× B) = 45 , then number of possible relations

in A → Bis

45 45 45 45 C 0 + C 1 + ... + C 45 = 2

Function

Definition:

A relation from a set A to a set B is called a function if

(i) Each element of set ‘A’ is associated with some

element in set ‘B’.

(ii) Each element of set ‘A’ has unique image in set ‘B’.

Ex.

∴ f ≡ {(1, a), (2, b), (3, c)}

So, it can be said that f ⊂ A × B.

Image and Pre-image

If an element (a ∈ A) is associated with an element

(1 ∈ B), then ‘1’ is called, the

“f image of a” or “image of a under f”

or

“the value of the function f at a”

or

“argument of a under the function f”.

Point to Remember!!!

If possible, n(A) = p; n(B) = q, then

number of possible relations in

A → B is pq

Function 4.

Ex.

f = {(a, 1), (b, 2), (c, 3), (d, 4)}

A = {a, b, c, d}, B = {1, 2, 3, 4, 5}

f: A → B

Ex.

A = {a, b, c, d}, B = {1, 2, 3, 4, 5}

f: A → B

(i) is a function.

Every element in A has a unique image in B.

(ii) is a function.

Every element in A has a unique image in B.

(iii) is not a function.

(‘d’ has no image in B)

(iv) is not a function.

(‘d’ does not have a unique image in B)

Function 6.

(ii) y = 2x – 1

Domain ≡ x ∈ R

M-1: For range, it can be seen from graph

∴ Range ≡ y ∈ R

M-2: y = 2x – 1

Since x ∈ (–∞, ∞)

y = 2(–∞, ∞) – 1

= (–∞, ∞) –

= (–∞, ∞).

∴ y ∈ R.

(iii) y = 3x + 4

Similarly,

Domain ≡ x ∈ R

Range ≡ y ∈ R

So, In general, for y = ax + b

(linear, a ≠ 0) ,

Domain ≡ x ∈ R

Range ≡ y ∈ R.

Solving Trick (for domain):

Try to think of values of x for which real

value of y does not exist.

Find the domain of

(i)

y x

Domain: x ∈ (-∞, 0) ∪ (0, ∞)

(at x = 0, y does not exit)

For range, draw the graph and check

especially for endpoints of intervals.

f(-∞) ≈ 0¯

f(0¯) = –∞

f(

) = +∞

f(∞) ≈ 0+

y ≠ 0

Q.

Q.

Q.

A.

A.

A.

7. Function

∴ Range: y ∈ R – {0}

Alternate Method:

x y 0 y

So, graph of

2 xy = 1 or xy = c is given:

(ii)

y 2x 1

Domain:

2x 1 0 or x R 2

For range,

y x YX Y y; X x 2 2 2 2

 −^  =^ ⇒^ =^  =^ =^ − 

So, using shifting of origin, it can be seen that

function is similar to xy = c 2

∴ Range: y ∈ R – {0}

(iii) y =

3x + 4

3x 4 0 x 3

∴ Domain:

x R 3

Similarly, range: y ∈ R – {0}

(iv) y = x

Since square root of a negative value is not real,

in case of y = f x ,( ) to find domain, make f(x)

Using concept, Domain of y = x is x ∈ [0, ∞).

From graph

Range: y ∈ [0, ∞)

Point to Remember!!!

Range of ( ) ( )

f x , a 0 ax b

will always be R– {0}.

Q.

Q.

Q.

A.

A.

A.

9. Function

Q.

Q.

A.

A.

(vii)

y x

Here, it is similar to

ax +b

So ax + b > 0

∴ Domain: x ∈ (0, ∞)

Range: y ∈ (0, ∞)

( ) (^ )^

 →^ →^ ∞

 ∞^ ∞ 

(viii)

y 3x 4

Similarly, Domain:

x , 3

Range: y ∈ (0, ∞) or R

Point to Remember!!!

ax + b, a ≠ 0 has its range

[0, ∞)

3x–

1

Algebraic operations on functions

(i) Let f and g be functions with domain D 1 and D 2

then the function f + g, is defined as

(f + g) (x) = f(x) + g(x); Domain: D 1

∩ D

2

In this case, both functions f(x) and g(x) must be

real simultaneously. Only then, the overall function

will be real.

(ii) Let f and g be function with domain D 1 and D 2

then the function f– g is defined as

(f – g) (x) = f(x) – g(x); Domain: D 1 ∩ D 2

Function 10.

Q.

A.

Again, both functions should be real at the same time. So, the domain

is set of all the values of x common to both of their domain.

(iii) Let f and g be functions with domain D 1 and D 2 , then the function ( )

f x g

is defined as

f x

g x

i.e., ( )

f f^ x x g g x

Domain:

D 1 (^) ∩ D 2 (^) : (^) { x | g (^) ( x (^) )≠ (^0) } or

D 1 (^) ∩ D 2 (^) − (^) { x | g (^) ( x )= (^0) }

In this case, Denominator, i.e., g(x) must not be zero.

Ex.

f(x) = x; g(x) = x^2 -

D f

= R; D

g

= R

f x x g x 1

x 2

• 1 ≠ 0 ⇒ x ≠ 1, –1.

∴ Domain of ( )

f x g

is x ∈ R – {1, –1}.

(iv) Let f and g be functions with domain D 1 and D2, then the function fg is

defined as

( fg^ ) ( x )^ = f^ ( x^ ). g^ ( x^ ) ;^ Domain:^ D 1^ ∩ D 2

f(x) = x 3

• 2x 2 and g(x) = 3x 2

• 1. Find domain of f ± g,fg and f/g.

D

f

= R; D

g

= R

D

f

∩ D

g

= R

(i) f ± g : Df ∩ Dg ⇒ Domain = R ∩ R = R

(ii) fg : Df ∩ Dg ⇒ Domain = R ∩ R = R

(iii) f/g : Df ∩ Dg ⇒ Denominator ≠ 0

1 x 3

∴ Domain:

R

Function 12.

Domain, Range and Graph of Trigonometric Functions

(i) y = sin x

Domain : x ∈ R; Range ∈ [–1, 1]

Maximum value of y = sin x is 1 at x. 2

π

Minimum value of y = sin x is –1 at

x. 2

π

Domain: x ∈ R

Range: y ∈ [–1, 1]

(ii) y = cos x

Domain : x ∈ R ; Range : [–1, 1]

Point to Remember!!!

y sin r

θ =

Point to Remember!!!

x cos r

θ =

13. Function

(iii) y= tan x

Domain: x R ( 2n 1 ) ;

π ∈ − + n ∈ I

Range: R

(iv) y = cot x

cosx cotx sinx

⇒ sin x ≠ 0

⇒ x ≠ nπ, n ∈ I

∴ Domain: x ∈ R – nπ, n ∈ I

Range: R

(v) y = cosec x

Graph can be constructed by observing the

graph of y = sin x.

Since

cosecx , sinx

= sin x ≠ 0

∴ x ≠ nπ

Domain: R – nπ

From graph it can be seen that

range ∈ (–∞, –1] ∪ [1, ∞)

15. Function

Q.

Q.

Q.

Q.

A.

A.

A.

A.

(ii) y = sin (x 2 )

x ∈ (- ∞ , ∞ )

⇒ x^2 ∈ [0, ∞ )

So, range is [-1, 1]

(iii) y = sin ( x )

x ∈ [0, ∞ )

Input for sin x is [0, ∞ )

∴ Range is [–1, 1]

(iv) y = cos 4 x^4 x sin 2 2

y =

2 x^2 x^2 x^2 x cos sin cos sin 2 2 2 2

 −^   + 

2 x^2 x cos sin 2 2

= cos

x 2· 2

= cos x.

Range of cos x is [–1, 1]

So, y ∈ [–1, 1]

(v) y = (sin x + 2)^2 + 1

sin x ∈ [–1, 1]

sin x + 2 ∈ [1, 3]

(sin x + 2)^2 ∈ [1,9]

y = (sin x + 2)^2 +1 ∈ [2, 10]

Function 16.

Q.

A.

(vi) y = 4 tan x cos x

There is loss of domain because of presence of tanx.

Due to presence of tanx, cos x ≠ 0

y = 4 tanx cosx = 4

sinx .cosx cosx

; (cosx ≠ 0)

y = 4 sinx (cosx ≠ 0 so sinx ≠ -1 and 1)

∴ y ∈ (–4, 4) (⸪ sin x ∈ (-1, 1))

Definition of Polynomial Function

If a function f is defined by f(x) = a 0 x n

• a 1 x n-

a 2 x n-

• … + an-1x + an, where n is a non-negative

integer and a 0 , a 1 , a 2 , …, an are real numbers and

a 0 ≠ 0, then f is called a polynomial function of

degree n.

Ex. f(x) = 2 is zero degree polynomial

Ex.

f(x) = ax^2 + bx + c … Quadratic Polynomial

f(x) = ax^3 + bx^2 + cx + d … Cubic Polynomial

Similarly, polynomial of degree 4 is called

‘Biquadratic’.

Point to Remember!!!

a 0 is called leading coefficient.

a n is called constant term.

f(0) = a n

Points to Remember!!!

(i) A polynomial function is always

continuous.

(ii) A polynomial of degree one is

called a linear function.

(iii) A polynomial of degree one with

no constant term is called an

odd linear function. y = mx is an

odd linear function.

(iv) A polynomial of degree odd has

its range ‘R’ but a polynomial of

degree even has a range which

is always a subset of R.

In case of polynomial, the term

with highest degree is always

dominant when compared to

other terms, whenever x=± ∞

(other terms are negligible

compared to it at x = ± ∞ ).

So, in case of odd degree

polynomial, its range is R.

In case of even degree polynomial,

the range is only a subset of R.

Function 18.

Q.

A.

Ex. ( )

2 f x = x + 1

Domain : x 2

• 1 ≥ 0

Since x^2 ∈ [0, ∞ ), x^2 + 1 ∈ [1, ∞ ) So, x^2 + 1 ≥ 0 is true for all real x.

∴ Domain: (– ∞ , ∞ )

Range: [1, ∞ )

f(x) =

2 x + ax + 4

(a) Find ‘a’ if range is [2, ∞ ).

(b) Find ‘a’ if domain is all real.

(a) Since,

2 x + ax + 4 ∈ [2, ∞ ).

x 2

• ax + 4 ∈ [4, ∞ ).

∴ Minimum value of x 2

• ax + 4 = 4

D

4a

2 16 a 4 4

= ⇒ a = 0

(b) Domain is all real.

it means x^2 + ax + 4 ≥ 0 ∀ x ∈ R.

∴ D ≤ 0

a 2

• 16 ≤ 0

a ∈ [–4, 4]

Definition of Fractional / Rational Function

A fractional function is of the form y = f(x) =

g x

h x

where g(x) and h(x) are polynomial and h(x) ≠ 0.

The domain of f(x) is set of real x such that h(x) ≠ 0.

Ex.

f(x)=

4 2

2

2x x 1

x 4

; D = {x|x ≠ ± 2}

Here, denominator ≠ 0

x 2

• 4 ≠ 0

x ≠ 2, -

19. Function

Definition of Exponential Function

A function f(x) = a x (a > 0, a ≠ 1, x ∈ R) is

called an exponential function.

Ex.

y = 2 x

x - ∞ -2 -1 0 1 2 ∞

y 0 +^

As x increases, y increases.

As x decreases, y decreases.

as x → - ∞ , y =

×

×

×…=O+

So, if a x > a y ⇒ x > y

Ex.

y =

x 1

2

X - ∞ -2 -1 0 2 ∞

Y ∞ 4 2 1 1/4 0 +

As x increase, y decreases

So, if a x > a y ⇒ x < y

Domain: (–∞, ∞)

Range: (0, ∞ )

Point to Remember!!!

Graph of a x

base a > 1

Domain: x ∈ (- ∞ , ∞ )

Point to Remember!!!

Graph of ax^ :

base 0 < a < 1