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MATHEMATICS
REVISION NOTES
IIT-JEE MAIN / MHTCET / CBSE /ICE

Functions Part-I
Last Minute Revision Notes For JEE Main
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Functions - [A]
FUNCTIONS

1
...
For example, let us imagine a spherical
rubber balloon into which air is being pumped
...
In
mathematics, we say that r is a function of time t and symbolically, it may be written,
r = f (t) : r is a function of time t
...
Hence we can write
V = g (t) : V is function of time t
...

In general, if the values of a variable y depend on the values of another variable x, we write
y = f (x) i
...
, y is a function of x
...

Here f (x) can be understood as an operator between x and y
...


Consider y = f (x) = x + 1

i
...


x (input)

0

–3

1

y (output)

1

–2

2

Section 1

1

Mahayodha Academy

Function - A

Ordered Pair : The combination of an input and output is called an ordered pair
...
, which satisfy the function:

1
...
It is formed by plotting ordered
pairs that satisfy function
...
Graph of y  f (x) is shown :

y = f (x) is a linear polynomial whose graph is a straight line
...
So to draw the graph of linear polynomials we needed to plot
only two ordered pairs and join them
...

2
...
3

Suppose it is given that y is the square of x
...

If acceleration is 2 m/s2, we can write, velocity (v) as a function of t
...
e
v (t) = 0 + 2t { Using v = u + at }

Intervals and Notations
To express values a variable can take, we use following notations
...

a< x  b
2

Section 1

or

x  (a, b]

Mahayodha Academy
a < x
or

Function - A

x  [a, b)

(iv) Infinite intervals :
If x can take all real values, than we can write as
–x  (– , ) or x  R

(v)

Other Notations :





aax< b
x
x  (a, )
x  [a, )
x  (– , b)
x  (– , b]

(vi) If x can take specific values, say x = a, x = b and x = c, then we can write x  {a, b, c}

1
...
One such restriction is that we
can never divide by zero (0)
...

Domain of y = f (x) is collection of all inputs that operator can take so that output of operator exists

OR
The set of values of x for which y takes real values (so that the function is well defined) is known as
the domain for that function
...


Illustrating the Concept :
Find the domain of the following functions
...

(ii)

y = 2 sin x
Trigonometric function sin x is defined for all value of x


x  R i
...
x  (– ,)
Section 1

3

Mahayodha Academy

Function - A
(iii) As denominator cannot be zero, x can not be 2

x  (– , 2)  (2, ) i
...
x  R – {2}
(iv) Square root of a negative number is not defined



3  x is defined if 3 – x  0
x 3


...
(ii)

Combining (i) and (ii), domain is :
(v)

x  (0, 1)  (1, 3]
f (x) is defined if : x + | x |  0

x  –x





x>0


 Using | x | 


 x if x  0  


  x if x  0  

Hence domain is x  (0, )
(vi) f (x) is defined if
1 – log10 x  0 and




1
...


x>0

log10 x  1 and x > 0
x  10
and x > 0
domain is x  (0, 10]

Range
Range of y = f (x) is collection of all outputs {f (x)} corresponding to each real input in the
domain
...


NATURE OF A FUNCTION

2
...


4

Section 2

Mahayodha Academy

Function - A

Note : As an even function satisfies, f   x   f  x  , f (x) possesses same value for values of x which are equal in
magnitude and opposite in signs
...

Therefore graph of an even function is symmetrical about y-axis i
...
left half is mirror image of right half and
right half is mirror image of left half, considering y-axis as mirror
...

Note : Graph of f  x   x 2 is symmetrical about y-axis
...

Note : Graph of f (x) = cos x is symmetrical about y-axis
...
2

Odd function
If a function y = f (x) satisfies f   x    f  x  for all values of x in its domain, then y = f (x) is called an odd
function
...

For example : f  1   f 1 , f  2    f  2  , f  3   f  3 ,
...
e
...


Illustrating the concept :
(i)

Consider f  x   x3
3

f   x     x    x3



f (x) satisfies, f   x    f  x 

Hence f  x   x3 is an odd function
...

Note : As graph is symmetrical about origin, left half of the graph can also be drawn by taking reflection of right half
in both x-axis as well as y-axis :

2
...
4

Function - A

(iv)

odd  odd = even

(i)

if f (x) + f (– x) = 0

(ii)

if f (x) – f (– x) = 0  f is even function

 f is odd function

The square of an even or an odd function is always an even function
...

The smallest value of T is called the period of the function
...
In case T is not independent of x, f (x)
is not a periodic function
...

For example :
Graph of f  x  = A sin x repeats after an interview of 2
...


Standard Results on Periodic Functions
1
...


secn x, cosecn x
tan n x , cot n x

3
...


x   x    x

5
...
These function are not periodic

Properties of Periodic Function
(i)

If f (x) has period T , then
(a)
cf (x) has also period T
(b)
f (x + c) is periodic with period T
Section 2

7

Mahayodha Academy

Function - A
(c)
(d)
(e)
(f)
(ii)

f (x)  c is periodic with period T
If constant is added, subtracted, multiplied or divided in periodic function, period remains
same
...

Inverse of a periodic function does not exist
...


If f (x) is periodic with period T, then
T
k f (cx + d) has period c , hence period is affected by coefficient of x only
...


For Example : sin (x – [x]) = sin ({x}) is periodic with period 1 as x –[x] is periodic with period 1
...

2
...


(c)

LCM of rational with irrational is not possible

kind

For Example : LCM of (2, 2, 6) is not possible as 2, 6 Irrational and 2  rational
...

1
...


Function - A

Transcendental functions :
Trancendental functions can be of the following types :
(a)
Logarithmic function (b)
Exponential function
(c)
Trigonometric function (d)
Inverse Trigonometric function

3
...


3
...

For example :
f  x   3x 4 is a monomial function of degree 4
...


(b)

Polynomial function :
A function f (x) = a0xn + a1xn  1 +… + an ,

(a0  0)

where a0, a1, a2,
...


Section 3

9

Mahayodha Academy

Function - A
For example :

(c)

f (x) = x1920 + 5x1919 + 6x

(polynomial of degree 1920)

g (x) = x2 + 3x + 3

(polynomial of degree 2)

h (x) = 7 = 7x0

(polynomial of degree 0)

(i)

Constant function : If degree of a polynomial function is 0, then polynomial function is called
as constant function
...


Rational algebraic function : A function of the form f (x) =

p ( x)
, where p(x) and q(x) are
q ( x)

polynomials and q(x)  0, is called a rational function
...

(d)

Irrational function : An algebraic function or rational function containing one or more radicals (nonintegral rational powers of x) is called an irrational function
...
g
...
2

x3  x
, 3x3 – x3/2 + 9x – 1
2 x2  9

Transcendental function
(a)

Logarithmic Function
Logarithmic function is represented as
y = loga x

where x > 0, a  (0, 1) (1, )

If a > 1, y increases as x increases (as seen from graph)
...


Continuity :
The graph of f (x) = loga x is continuous (i
...
no break in the curve) in the respective domain
...


Properties of Logarithmic Function :
(i)
10

loga a = 1
Section 3

Mahayodha Academy
(ii)

Function - A

loga 1 = 0

(iii) loga (mn) = loga m + loga n
(iv) loga (m/n) = loga m – loga n
(v)

loga xm = m loga x

(vi)

log a b 

log mb
m  1, m > 0
log m a

(vii) a log a x  x
 x  y , if m  1

(ix) If logm x > logm y  

 x  y , if 0  m  1

a  mb , if m  1
(xi) logm a > b  
b
a  m , if 0  m  1

(b)

(viii)

alogc b = blogc a

(x)

logm a = b  a = mb

(xii)

 a  mb , if

logm a < b  
 a  mb , if

m 1
0  m 1

Exponential Function
y = ax where a > 1 or 0 < a < 1 is an exponential function of x
...
e
...

As observed from the graph, if a > 1, then y increases as x increases
...


Continuity :
The graph of f (x) = ax is continuous (i
...
no break in the curve)
everywhere
...


(c)

Trigonometric Functions
(i)

y = A sin (mx)

(ii)

y = A cos (mx)

(iii)

y = A tan (mx)

Period :
Period of y = A sin (mx) and
y = A cos (mx) is T = (2)/m
Period of y = A tan (mx) is T = /m
Section 3

11

Mahayodha Academy

Function - A

Continuity :
The graph of y = A sin (mx) and y = A cos (mx) is continuous
(i
...
no break in the curve) every where
...
e
...

The Domain of y = A tan (mx) is
x  R   2n  1

(iv)


and Range is y  R
...

m

y = A sec mx

Period :
It is a periodic function with period = 2/m
...


12

Section 3

Mahayodha Academy
(vi)

Function - A

y = A cosec mx

Period :
It is a periodic function with period = 2/m
...


Domain and Range :
The domain of y = A cosec mx is R – {n /m} and the
range is y     ,  A   A,  

(d)

Inverse Trigonometric Function :
(To be discussed in the later module of Function)
...
3

Piecewise Defined functions :
(a)

Modulus or Absolute Value Function
Modulus function is a numerical value function or we also call it as absolute value function
...
53  2
...
etc
...

For example, if x = 1 means distance covered is one unit on right hand side or left side of origin
...

The expression x can be further split as follows :
x ; x  0
y= x = 
 x ; x  0

Continuity :
The graph of y = x is continuous (i
...
no break in the curve) but has a corner at origin as shown
...


Results :
(A) If a > 0 then;
(i)

| f (x) | = a



f (x) = + a

(ii)

| f (x) | < a



– a < f (x) < a

(iii)

| f (x) | > a



f (x) < – a

or

f (x) > a

(B) If a < 0 then ;
(i)

| f (x) | = a



no solution

(ii)

| f (x) | < a



no solution

(iii)

| f (x) | > a



all real values of x in domain of f (x)

(b) Greatest integer function (unit step function) :
y = [ x ] = the greatest integer less than or equal to x
...


Continuity :
The graph of f (x) is discontinuous (i
...
break in the curve) at integral values of x
...


14

Section 3

Mahayodha Academy
Note :

Function - A

(a)

[ x ] = x holds if x is integer
...


(c)

x = [x] + {x}, {x} denotes the fractional part of x
...

We can extend graph for other values of x
...
Hence y = {x} is discontinuous
 x I
...
Therefore y = {x}is a periodic function with
period 1
...


(e)

Signum Function :
1 , x  0

y = sgn (x) = 0 , x  0
 1 , x  0


Section 3

15

Mahayodha Academy

Function - A
This can also be written as

y = sgn (x) =

x

x

0

, x0
, x0

Domain and Range
Domain of y = sgn (x) is x  R and range is y  1, 0, 1

Continuity
The graph of f (x) is continuous for all values of x except at x = 0 where there is a break in the curve
...
There are other
graphs that you should learn and remember
...


16

1
...


y=–x

3
...


y=–|x|

5
...


–|y|=x

Section 4

Mahayodha Academy

Function - A

7
...


y2 = – 4ax

9
...


x2 = – 4ay

11
...
y  n , n is odd integer  1
x

greater is ‘a’ steeper is curve
...
e
...
can be
x x3 x5

considered same (for rough sketching)]
...
y  n , n is even integer > 1
x

[i
...
graph of y 

1
1 1
,
,
,
...


14
...
e
...
can be
considered same (for rough sketching)]
...
y  xn , n is odd integer > 1

[i
...
graph of y  x3 , x5 , x7 ,
...


TRANSFORMATIONS

5
...


Section - 5

Transformations
(i)

Transformation 1

(a)

y = f (x) 
 y = f (x + a)
To draw y = f (x + a), shift the graph of
y = f (x) through ‘a’ units towards left
...


Illustrating the concept :
Plot the graph of following :

18

(i)

4y = 2x

(i)

4y =2x  y = 2x – 2
...
2x
(ii)

y = 4
...


(b)

y = f (x) 
 y  a = f (x)
To draw y  a = f (x), shift the
graph of y = f (x) by a units upward
...


1  cos 2 x

2

y=

1 cos 2 x 
cos 2 x
+
y=
y

a

f
(
x
)
2
2
2

Shift 1/2 unit up

Observation:
In f (x) = cos2 x maximum value of f (x) is 1 and minimum value of f (x) is 0
...


Illustrating the concept :
Sketch the graph of following function : (i)
(i)

20

 y = ex
y = e–x 
f ( x)  f ( x)

Section 5

y –e–x = 0 , (ii)

 π π
y = cosec (x)  x   – 2 , 2 



Mahayodha Academy
(ii)

Function - A

 y = cosec (x)
y = cosec (–x) 
f ( x)  f ( x)

(iv) Transformation 4
y = f (x) 
 y =  f (x)
To plot y =  f (x), Draw y = f (x) first and then take the mirror image of y = f (x) in x-axis
...
Also include the
right portion of the graph of y = f (x)
...


Illustrating the concept :
Draw the graph of the following curves :
(i)

y = | x2 – 2x – 3| ;

(i)

2

y = | x2 – 2x – 3 | 
f ( x)  | f ( x)| y = x – 2x – 3

(ii)

y = | log x |
...
Also include the upper portion of the curve y = f (x)
...

y = f (x) 

To plot y = [ f (x) ] use the following working rule :
(i)
(ii)

Draw y = f (x)
...

From each intersection point draw horizontal lines upto nearest right vertical line such
that the horizontal line is always below the graph
...

Let f : A  B and g : B  C be two functions then gof : A  C
...

Note :
(i) fog(x) = f (g (x)) ;

(ii)

fof (x) = f (f (x)) ;

(iii) gog(x) = g(g(x)) ;

(iv)

gof (x) = g (f (x))
...
But for IInd block, g(x) i
...
the dependent variable of Ist block is independent variable corresponding
fog(x) is the dependent variable
...


As

(fog) (x) = f {g (x)} = f (x2) = sin x2


...
(ii)

(fog) (x)  (gof) (x)

From (i) and (ii),
(ii)

Find the values of (fog)

From (i),

  
1
 fog  x  sin x 2   fog   2   sin    






4

2

2

2    3   3

sin

3  2 
4
3


and From (ii),  gof  x  sin2 x   gof  

Illustration - 1
(A)

The period of the following functions f (x) = sin4 x  cos 4 x is :
(B)

2



(C)


2

(D)


4

SOLUTION : (C)
f (x) = sin4 x  cos 4 x

2
=  sin2 x  cos 2 x   2sin2 x cos 2 x
= 1

1
1
2
 2 sin x cos x   1  sin2 2 x
2
2
[Using sin 2 = 2 sin cos)

1
3 1
1  cos 4 x    cos 4 x

4
4 4
2 

Period =
4
2
= 1

Section 6

27

Mahayodha Academy

Function - A

Illustration - 2
(A)

2
k

The period of the functions f (x) = a sin kx + b cos kx is :
(B)


k


2k

(C)

(D)

None of these

SOLUTION : (A)
f (x) = a sin kx + b cos kx
a



f  x   a2  b2 

 a2  b2

sin kx 

b
a 2  b2



cos kx 


 a 2  b2 sin  kx    , where   tan1 b
a

Period of f  x  

2
k

Another Approach :
We know, period of sin kx 

2
2
and period of coskx 
k
k

 2 2 
2
,
Hence period of f  x   a sin kx  b cos kx = LCM of 
 =
k
k 
 k

Illustration - 3
If f  x  
(A)

2

(B)


sin 2nx
, n  N has
as it’s fundamental period
...
=
6
1  cos 2 nx
3
(C)
6
(D)
12

SOLUTION : (C)

f  x 

sin 2nx
1  cos 2nx 
1 

2



2 sin 2nx
3  cos 2nx
2 

2n n

T1 :

period of sin 2nx 

T2 :

period of cos 2nx  2  
2n n

Illustration - 4

Hence period of f  x  


n

But

period of f (x) =


(given)
6



n=6

If f  x   sin    x  is a periodic function with period  where [
...
(i)
x 2  1  0   x  1  x  1  0 

x 1, 1


...
(i)

For f ( x )  loga x is defined for
 x  0, a  0, a  1
 3 x 
log10 
 is defined for
 x 

3 x
0
x

 0 x3


...

2


30

Section 6



Mahayodha Academy
Illustration - 9

Function - A

The domain of the function f  x  

(A)

  ,  2 4, 

(B)

  ,  2  4, 

(C)

  ,  2 4, 

(D)

None of these

1

 x 2   x   6

is x  

SOLUTION : (A)



f  x  is defined for

x  2

Also  x  3 for  x   4, 5, 6
...

But  x  2 for  x   3,  4,  5,
...


1  4 y 2  0, y  0



Domain of y is x  ,
...


above quadratic equation  0

Illustration - 11

 2 y  1  2 y   0, y  0

 1 1
Range of y    , 
 2 2

x2
is y 
1  x2

(0, 1) (C) [0, 1]

(D)

None of these

Section 6

31

Mahayodha Academy

Function - A
SOLUTION : (A)
y is defined for all real x
...
e
...

(A)

0

(B)

1

(C) 2

(D)

None of these

SOLUTION : (A)
f  x  2  f  x  4

As f  x  is an odd function




As

Using (ii) and (iii) we hae :

f  x    f  x 

Put x  0



f  x   f  x  4

f  0    f  0

Put x  0


...
(iii)

f 0  f  0  4

But f  0  0


...
[Using (i)]

Hence value of f  4  0

Replace x by x  2

Illustration - 13
The domain of the definition of the function f  x  
integer less than or equal to x is :
(A) R
(B)
0,  

(C)

  , 0 

(D)

2

x 2   x 2  , where [x] is greatest
 
None of these

SOLUTION (D) :
For f (x) to take real values, expression inside
square root should be non-negative
...
e
...
From graaph it is obvious that
difference of x 2 and x 2 is non-negative for all

 

x  0,   i
...
region right of y-axis as well as

negative integers
...


The domain of the function f  x   log x cos x  x    ,   is x 

   
 2 , 2   1



(B)

 
 2



,   1
2

(C)





  
,
2 2 

(D)

None of these

SOLUTION : (D)

log g  x  f  x  is defined for

f  x   0 and

cos x > 0 and



x   0, 1  1,  


...
(ii)

Domain of f (x) is x   0,


  - {1}
2 

 2

The range of the function is : f  x   log e  3x  4 x  5 

11 
3 

(B)


 log e


11 
, 
3



11
11
(C)   log e 3 , log e 3 



(D) None of these

SOLUTION : (B)
 D

Range of 3x 2  4 x  5 is  4a ,  


Where D is the discriminant and a is coeffi
cient of x2 of quadratic expression
3x 2  4 x  5



Range of

  16  60 

3x 2  4 x  5  






, 



43

11

Range of 3x 2  4 x  5  

3



, 


From graph of y  log e t , we can observe that

Section 6

33

Mahayodha Academy

Function - A

11

  3x 2  4 x  5    ,  

 3





11 
log  3x 2  4 x  5   log e ,  

 
3


11

 t   ,  ,
3



Illustration - 17

(A)


11 
 ,


3 


 2

Hence range of log  3x  4 x  5  is


11 
log e t  log e ,  
3



[loge 11/3,  )

The range of the function is : f  x   3x 2  4 x  5 is :
(B)


11 
  ,


3 


(C)

 11

,  

 3


(D)

 11

,  

 3



SOLUTION : (C)
 D

Range of 3x 2  4 x  5 is  4a ,  



Where D is the discriminant and a is coefficient of
x2 of quadratic expression 3x 2  4 x  5


Range of

  16  60 


,


2
3x  4 x  5  4  3






11

Range of is 3x 2  4 x  5 is  3 ,  



From graph of, we can observe that
 11

11

,  
 t   , , t  
3

 3


34

Section 6



11

  3x 2  4 x  5   




3

 11

3x 2  4 x  5  



, 

 3





,  



Hence range of is 3x 2  4 x  5 is
 11

,  

 3


Mahayodha Academy

Function - A

IN - CHAPTER EXERCISE
1
...


y  x2  2x  3

y  1 e

x

 e x

(where [
...


Intervals and Notations
Interval

2
...


3
...


4
...


Nature of a function
Even function

f (x) = f (x)

eg : f (x) = x2, cos x

Odd function

f (x) =  f (x)

eg : f (x) = x3, sin x

Properties of Even and Odd function :
(a) Sum :

(i)
(ii)
(iii)
(iv)

(b) Difference : (i)
(ii)
36

Things to Remember

even + even = even
even + odd = neither even nor odd
odd + even = neither even nor odd
odd + odd = odd
even – even = even
even – odd = neither even nor odd

Mahayodha Academy
(iii)
(iv)

odd – even = neither even nor odd
odd – odd = odd

(c)

Product :

(i)
(ii)
(iii)
(iv)

even × even = even
even × odd = odd
odd × even = odd
odd × odd = even

(d)

Division :

(i)
(ii)
(iii)
(iv)

even  even = even
even  odd = odd
odd  even = odd
odd  odd = even

(i)

if f (x) + f (– x) = 0

(ii)

if f (x) – f (– x) = 0  f is even function

(e)

(f)

6
...


Periodic function
A function f (x) is said to be periodic function of x, if there exists a positive real number T such that
f (x+ T) = f (x), for all values of x in the domain of f (x)
...


7
...
sin n x, cos n x
secn x, cosecn x

Period
, If n is even
2, If n is odd or fraction

2
...
| sinx |, | cos x |
| tan x |, | cot x |
| sec x |, | cosec x |



4
...


8
...


1
These functions are not periodic

Properties of Periodic Function
(i)

If f (x) has period T , then
Things to Remember

37

Mahayodha Academy

Function - A
(a)
(b)
(c)
(d)
(e)
(f)

cf (x) has period T also
f (x + c) is periodic with period T
f (x)  c is periodic with period T
If constant is added, subtracted, multiplied or divided in periodic function, period remains
same
...

Inverse of a periodic function does not exist
...


(iii)

If f (x) is periodic with period T, then
T
k f (cx + d) has period c , hence period is affected by coefficient of x only
...


It is a function of the form y  f  x   b , where b is a constant
...
That is why this function is called
an identify function
...
So domain and range of the
identify function is all real values of x i
...
, x  R and y  R
...
e
...


10
...


Domain

Range

x  R Depends upon k and n

38

Things to Remember

Continuity

Periodic

always continuous

not periodic

Mahayodha Academy

Function - A

Function
Polynomial function

Rational function

Definition
A function f (x) = a0x + a1xn  1 +… + an , (a0  0)
where a0, a1, a2,
...

n

Domain

Range

Continuity

xR

depends

always continuous

A function of the form f (x) =

Periodic
not periodic

p ( x)
, where p(x) and q(x) are
q ( x)

polynomial and q (x)  0, is called rational function
...

Domain
xR

 {q (x) = 0}

Range

Continuity

depends

discontinuous at not
x q x   0

(ii)

Periodic

Periodic
periodic

Transcendental functions
Function
Logarithmic

Definition
Logarithmic function is represented as y = loga x where x > 0,

function

a  (0, 1) (1, )
If a > 1, y increases as x increases, if 0 < a < 1, y decreases as x
increases
...
e
...

If a > 1, then y increases as x increases, if 0 < a < 1, then y decreases
as x increases
...

The expression x can be further split as follows :
x ; x  0
y= x = 
 x ; x  0
Domain

Range

Continuity

Periodic

xR

y0

always continuous

not periodic

Definition
y = [ x ] = the greatest integer less than or equal to x
...

Domain

Range

Continuity

xR

yI

discontinuous at not periodic
xI

40

Things to Remember

Periodic

Mahayodha Academy

Function - A

Function

Definition

Fractional part

Fractional part function is represented as y   x  x   x
Domain

Range

Continuity

Periodic

xR

y   0, 1

discontinuous at

periodic with
period 1

xI

Function

Definition

1 , x  0

0 , x0
y = sgn (x) = 
1 , x  0


Signum

Domain

Range

Continuity

Periodic

xR

1, 0, 1

discontinuous at
f (x) = 0

not periodic

11
...
No
...
(a)

y = f (x) 
 y = f (x + a)

Shift the graph of y = f (x) through ‘a’ units towards left
...


2
...


(b)

y = f (x) 
 y  a = f (x)

Shift the graph of y = f (x) by ‘a’ unit upward
...


y = f (x) 
 y = f (x)

Draw y = f (x) first then take the mirror image of y = f (x) in the y
axis
...


y = f (x) 
 y =  f (x)

Draw y = f (x) first and then take the mirror image of y = f (x) in xaxis
...


y = f (x) 
 y = f (| x |)

Draw the graph of y = f (x) first, then remove the left portion of the
graph after that take the mirror image of the right portion of the
curve in the Y-axis
...


6
...


y = f (x) 
 | y | = f (x)

Draw y = f (x) first then remove the lower portion of the curve and
then take the mirror image of upper portion of the curve in xaxis
...


8
...


(i)

Draw y = f (x)
...


(iv)

From each intersection point draw horizontal lines upto
nearest right vertical line such that the horizontal line is always
below the graph
...


Composite functions
Let A, B and C be three non-empty sets
...
This function is called Composite
Function
...


Things to Remember

Mahayodha Academy

Function - A

SOLUTION - IN - CHAPTER EXERCISE
1
...
(iii)


(iii)

y |y|

| y | = | log | x ||




...
(ii)

x+y=1


...
(iii)

 f(x)  | f (x)|

...
(i)

x|x|

(iv)

y  2  x2

Solutions

43

Mahayodha Academy

Function - A
(v)

y = [sin[x]]

...
(iv)

y|y|

Solutions

y = 1 + e| x | – e–x


...
(i)

0  x 1

1  x2

2
...
(ii)

x  [x]


...
(i)

Mahayodha Academy
1  | x  1 |  | x  1|  1
f  x  1  
0
 | x  1|  1

x0
 0
 x
0 x  1


2  x 1  x  2
 0
x2
1  | x  1|  | x  1|  1
f  x  1  
0
 | x  1|  1

x2
 0
2  x 2  x   1


1 x  0
 x
 0
x0

Function - A


...
(2)

Combining graph (1) and (2)

Solutions

45



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