Important Type of Functions JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Important Type of Functions JEE Notes | EduRev

 Page 1


 
 
C. IMPORTANT TYPE OF FUNCTIONS
(1) Trigonometric function :
Function Domain Range         Curve
     (i) f(x) = sin x          x ? R y ? [–1, 1]   x
y = sin x
1
–1
2 / 3 ?
2 / ? ? 2
? ?2 ? ? 2 / 3 ? ? ?
2 / ? ?
     (ii) f(x) = cos x x ? R y ? [–1, 1]     
x
y = cos x
1
–1
2 / 3 ?
2 / ? ? 2
? ?2
? ?
2 / 3 ? ?
?
2 / ? ?
2 / 5 ?
     (iii) f(x) = tan x x ? R – (2n + 1) 
2
?
, n ? ? ? y ? R       
y = tan x
x
O
2
? ?
2
3 ?
2
3 ?
? ? ?
2
?
?
     (iv) f(x) = cot x x ? R – n ? ? n ? ? ? y ? R
y = cot x
x
O
2
?
?
2
3 ?
? ?
2
?
?
? 2
     (v) f(x) = cosec x x ? R –n ?, n ? ? ? ? ? ??? ? y ? ?? ? ?, –1] ? ?[1, ?)     
1
y = cosec x
2 / ?
0
–1
2 / ? ?
? ?
2 / 3 ?
? 2
x
?
Page 2


 
 
C. IMPORTANT TYPE OF FUNCTIONS
(1) Trigonometric function :
Function Domain Range         Curve
     (i) f(x) = sin x          x ? R y ? [–1, 1]   x
y = sin x
1
–1
2 / 3 ?
2 / ? ? 2
? ?2 ? ? 2 / 3 ? ? ?
2 / ? ?
     (ii) f(x) = cos x x ? R y ? [–1, 1]     
x
y = cos x
1
–1
2 / 3 ?
2 / ? ? 2
? ?2
? ?
2 / 3 ? ?
?
2 / ? ?
2 / 5 ?
     (iii) f(x) = tan x x ? R – (2n + 1) 
2
?
, n ? ? ? y ? R       
y = tan x
x
O
2
? ?
2
3 ?
2
3 ?
? ? ?
2
?
?
     (iv) f(x) = cot x x ? R – n ? ? n ? ? ? y ? R
y = cot x
x
O
2
?
?
2
3 ?
? ?
2
?
?
? 2
     (v) f(x) = cosec x x ? R –n ?, n ? ? ? ? ? ??? ? y ? ?? ? ?, –1] ? ?[1, ?)     
1
y = cosec x
2 / ?
0
–1
2 / ? ?
? ?
2 / 3 ?
? 2
x
?
     (vi) f(x) = sec x   x ?R – (2n+1) 
2
?
, n ? ?   y ? ?? ? ?, –1] ? ?[1, ?)  
1
y = sec x
0
–1
2 / ? ? 2 / 3 ?
x
?
2 / ?
? ?
2 / 3 ? ?
(2) Polynomial Function :
f(x) = a
0
x
n
 + a
1
x
n – 1
 + a
2
 x
n–2
 +.......+ a
n
where a
0
, a
1
, a
2
 ........ a
n
 ? R n ? W
If a
0
 ? 0, then f(x) is called n
th
 degree polynomial and Domain x ? R
(3) Algebraic Function : A function is called an algebraic function. If it can be constructed using
algebraic operations such as additions, subtractions, multiplication, division taking roots etc.
All polynomial functions are algebraic but converse is not true.
Ex : f(x) = 
2 4
x 5 x ?
 + x + (x
3
 + 5)
3/5
, f(x) = x
3
 + 3x
2
 + x + 5
Remark : Function which are not algebraic are called as TRANSCENDENTAL FUNCTION.
Ex : f(x) = 
3
5 / 3 2 5
x
) x 5 x ( ?
 + 3 6 x 5 x
2
? ? + ?n x  ? transcidental function
Ex : f(x) = 7 x
2
? + e
?n x
 + 
7 x
7 x
2
?
?
  ? algebraic function.
(4) Rational Function : It is a function of form f(x) = 
) x ( h
) x ( g
, where g(x) & h(x) are poly. function and
h(x) ? 0 Ex. f(x) = 
4 x
2 x 3 x
2
2 4
?
? ?
(5) Logarithmic function : f(x) = log
a
x, where x > 0, a > 0, a ? 1
a ? base, x ? number or argument of log.
Case–I : 0 < a < 1 Case–II : a > 1
             f(x) = log
a
 x    f(x) = ?n x
             Domain : x ? (0, ?)     
O
(1, 0)
x
f(x)
   
O (1, 0) x
f(x)
             Range : y ? R
(6) Exponential function : f(x) = a
x
, where a > 0, a ? 1
a ? Base x ? Exponent
Case–I : 0 < a < 1 ; a = 1/2 Case–II : a > 1
             f(x) = 
x
1
2
? ?
? ?
? ?
    
O
(1, 0)
x
f(x)
O
(0,1)
x
f(x)
             Domain : x ? R
             Range : y ? (0, ?)
Page 3


 
 
C. IMPORTANT TYPE OF FUNCTIONS
(1) Trigonometric function :
Function Domain Range         Curve
     (i) f(x) = sin x          x ? R y ? [–1, 1]   x
y = sin x
1
–1
2 / 3 ?
2 / ? ? 2
? ?2 ? ? 2 / 3 ? ? ?
2 / ? ?
     (ii) f(x) = cos x x ? R y ? [–1, 1]     
x
y = cos x
1
–1
2 / 3 ?
2 / ? ? 2
? ?2
? ?
2 / 3 ? ?
?
2 / ? ?
2 / 5 ?
     (iii) f(x) = tan x x ? R – (2n + 1) 
2
?
, n ? ? ? y ? R       
y = tan x
x
O
2
? ?
2
3 ?
2
3 ?
? ? ?
2
?
?
     (iv) f(x) = cot x x ? R – n ? ? n ? ? ? y ? R
y = cot x
x
O
2
?
?
2
3 ?
? ?
2
?
?
? 2
     (v) f(x) = cosec x x ? R –n ?, n ? ? ? ? ? ??? ? y ? ?? ? ?, –1] ? ?[1, ?)     
1
y = cosec x
2 / ?
0
–1
2 / ? ?
? ?
2 / 3 ?
? 2
x
?
     (vi) f(x) = sec x   x ?R – (2n+1) 
2
?
, n ? ?   y ? ?? ? ?, –1] ? ?[1, ?)  
1
y = sec x
0
–1
2 / ? ? 2 / 3 ?
x
?
2 / ?
? ?
2 / 3 ? ?
(2) Polynomial Function :
f(x) = a
0
x
n
 + a
1
x
n – 1
 + a
2
 x
n–2
 +.......+ a
n
where a
0
, a
1
, a
2
 ........ a
n
 ? R n ? W
If a
0
 ? 0, then f(x) is called n
th
 degree polynomial and Domain x ? R
(3) Algebraic Function : A function is called an algebraic function. If it can be constructed using
algebraic operations such as additions, subtractions, multiplication, division taking roots etc.
All polynomial functions are algebraic but converse is not true.
Ex : f(x) = 
2 4
x 5 x ?
 + x + (x
3
 + 5)
3/5
, f(x) = x
3
 + 3x
2
 + x + 5
Remark : Function which are not algebraic are called as TRANSCENDENTAL FUNCTION.
Ex : f(x) = 
3
5 / 3 2 5
x
) x 5 x ( ?
 + 3 6 x 5 x
2
? ? + ?n x  ? transcidental function
Ex : f(x) = 7 x
2
? + e
?n x
 + 
7 x
7 x
2
?
?
  ? algebraic function.
(4) Rational Function : It is a function of form f(x) = 
) x ( h
) x ( g
, where g(x) & h(x) are poly. function and
h(x) ? 0 Ex. f(x) = 
4 x
2 x 3 x
2
2 4
?
? ?
(5) Logarithmic function : f(x) = log
a
x, where x > 0, a > 0, a ? 1
a ? base, x ? number or argument of log.
Case–I : 0 < a < 1 Case–II : a > 1
             f(x) = log
a
 x    f(x) = ?n x
             Domain : x ? (0, ?)     
O
(1, 0)
x
f(x)
   
O (1, 0) x
f(x)
             Range : y ? R
(6) Exponential function : f(x) = a
x
, where a > 0, a ? 1
a ? Base x ? Exponent
Case–I : 0 < a < 1 ; a = 1/2 Case–II : a > 1
             f(x) = 
x
1
2
? ?
? ?
? ?
    
O
(1, 0)
x
f(x)
O
(0,1)
x
f(x)
             Domain : x ? R
             Range : y ? (0, ?)
(7) Absolute value function (Modulus function) :
y = |x| = 
?
?
?
? ?
?
0 x ; x
0 x ; x
y   x = y=–x
y
x
Domain : x ? R; Range : y ? R
+
 ? {0}
(8) Signum function :
y = sgn (x) = 
?
?
?
?
?
? ?
?
?
0 x ; 1
0 x ; 0
0 x ; 1
   
y
1
0
–1
x
Domain : x ? R; Range : y ? {–1, 0, 1}
(9) Greatest integer function (step-up function) :
y = f(x) = [x] 
?
?
?
?
? ?
x than less
otherwise ; Integer Greatest
I x ; x
1 2
–1
0
1
2
y
3 x
Domain : x ? R; Range : y ? I
Ex : [2
.
3] = 2, [5] = 5, [–2
.
3] = –3
Properties :
(i) [x] ? x < [x] + 1 (ii) [x + m] = [x] + m ; m ? I (iii) [x] + [–x] = 
?
?
?
? ?
?
I
I
x ; 1
x ; 0
(10) Fractional part function :
y = f(x) = {x} = x – [x]
Domain : x ? R; Range : [0, 1)
Ex : 2.3 = 2 + 0.3 ? ?fractional part
1 –1 2 –2 0
y
3 x
1
               ?
        Integer part
Properties :
(i) Fractional part of any integer is zero. (ii) {x + n} = {x}, n ? ? ?
(iii) {x} + {–x} = 
?
I 0 ; x
1 ; otherwise
?
Ex.6 Find the range of the following functions : (a) y = 
x 3 cos x 3 sin 2
1
? ?
  (b) y = sin
–1
 
?
?
?
?
?
?
?
?
?
?
2 x
1 x
2
2
Sol. (a) We have y = 
x 3 cos x 3 sin 2
1
? ?
  i.e. sin 3x + cos 3x = 
y
1
 – 2
i.e.
2
 sin 
1
3x
4 y
? ? ?
? ?
? ?
? ?
 – 2 i.e. sin 
1
3x
4 2
? ? ?
? ?
? ?
? ?
?
?
?
?
?
?
?
?
? 2
y
1
since, 
?
?
?
?
?
? ?
?
4
x 3 sin
 ? 1, therefore we have
2
y
1
?
 ? 
2
i.e. –
y
1
2 ?
 – 2 ? 
2
i.e. 2 – 
y
1
2 ? ? 2 2
Page 4


 
 
C. IMPORTANT TYPE OF FUNCTIONS
(1) Trigonometric function :
Function Domain Range         Curve
     (i) f(x) = sin x          x ? R y ? [–1, 1]   x
y = sin x
1
–1
2 / 3 ?
2 / ? ? 2
? ?2 ? ? 2 / 3 ? ? ?
2 / ? ?
     (ii) f(x) = cos x x ? R y ? [–1, 1]     
x
y = cos x
1
–1
2 / 3 ?
2 / ? ? 2
? ?2
? ?
2 / 3 ? ?
?
2 / ? ?
2 / 5 ?
     (iii) f(x) = tan x x ? R – (2n + 1) 
2
?
, n ? ? ? y ? R       
y = tan x
x
O
2
? ?
2
3 ?
2
3 ?
? ? ?
2
?
?
     (iv) f(x) = cot x x ? R – n ? ? n ? ? ? y ? R
y = cot x
x
O
2
?
?
2
3 ?
? ?
2
?
?
? 2
     (v) f(x) = cosec x x ? R –n ?, n ? ? ? ? ? ??? ? y ? ?? ? ?, –1] ? ?[1, ?)     
1
y = cosec x
2 / ?
0
–1
2 / ? ?
? ?
2 / 3 ?
? 2
x
?
     (vi) f(x) = sec x   x ?R – (2n+1) 
2
?
, n ? ?   y ? ?? ? ?, –1] ? ?[1, ?)  
1
y = sec x
0
–1
2 / ? ? 2 / 3 ?
x
?
2 / ?
? ?
2 / 3 ? ?
(2) Polynomial Function :
f(x) = a
0
x
n
 + a
1
x
n – 1
 + a
2
 x
n–2
 +.......+ a
n
where a
0
, a
1
, a
2
 ........ a
n
 ? R n ? W
If a
0
 ? 0, then f(x) is called n
th
 degree polynomial and Domain x ? R
(3) Algebraic Function : A function is called an algebraic function. If it can be constructed using
algebraic operations such as additions, subtractions, multiplication, division taking roots etc.
All polynomial functions are algebraic but converse is not true.
Ex : f(x) = 
2 4
x 5 x ?
 + x + (x
3
 + 5)
3/5
, f(x) = x
3
 + 3x
2
 + x + 5
Remark : Function which are not algebraic are called as TRANSCENDENTAL FUNCTION.
Ex : f(x) = 
3
5 / 3 2 5
x
) x 5 x ( ?
 + 3 6 x 5 x
2
? ? + ?n x  ? transcidental function
Ex : f(x) = 7 x
2
? + e
?n x
 + 
7 x
7 x
2
?
?
  ? algebraic function.
(4) Rational Function : It is a function of form f(x) = 
) x ( h
) x ( g
, where g(x) & h(x) are poly. function and
h(x) ? 0 Ex. f(x) = 
4 x
2 x 3 x
2
2 4
?
? ?
(5) Logarithmic function : f(x) = log
a
x, where x > 0, a > 0, a ? 1
a ? base, x ? number or argument of log.
Case–I : 0 < a < 1 Case–II : a > 1
             f(x) = log
a
 x    f(x) = ?n x
             Domain : x ? (0, ?)     
O
(1, 0)
x
f(x)
   
O (1, 0) x
f(x)
             Range : y ? R
(6) Exponential function : f(x) = a
x
, where a > 0, a ? 1
a ? Base x ? Exponent
Case–I : 0 < a < 1 ; a = 1/2 Case–II : a > 1
             f(x) = 
x
1
2
? ?
? ?
? ?
    
O
(1, 0)
x
f(x)
O
(0,1)
x
f(x)
             Domain : x ? R
             Range : y ? (0, ?)
(7) Absolute value function (Modulus function) :
y = |x| = 
?
?
?
? ?
?
0 x ; x
0 x ; x
y   x = y=–x
y
x
Domain : x ? R; Range : y ? R
+
 ? {0}
(8) Signum function :
y = sgn (x) = 
?
?
?
?
?
? ?
?
?
0 x ; 1
0 x ; 0
0 x ; 1
   
y
1
0
–1
x
Domain : x ? R; Range : y ? {–1, 0, 1}
(9) Greatest integer function (step-up function) :
y = f(x) = [x] 
?
?
?
?
? ?
x than less
otherwise ; Integer Greatest
I x ; x
1 2
–1
0
1
2
y
3 x
Domain : x ? R; Range : y ? I
Ex : [2
.
3] = 2, [5] = 5, [–2
.
3] = –3
Properties :
(i) [x] ? x < [x] + 1 (ii) [x + m] = [x] + m ; m ? I (iii) [x] + [–x] = 
?
?
?
? ?
?
I
I
x ; 1
x ; 0
(10) Fractional part function :
y = f(x) = {x} = x – [x]
Domain : x ? R; Range : [0, 1)
Ex : 2.3 = 2 + 0.3 ? ?fractional part
1 –1 2 –2 0
y
3 x
1
               ?
        Integer part
Properties :
(i) Fractional part of any integer is zero. (ii) {x + n} = {x}, n ? ? ?
(iii) {x} + {–x} = 
?
I 0 ; x
1 ; otherwise
?
Ex.6 Find the range of the following functions : (a) y = 
x 3 cos x 3 sin 2
1
? ?
  (b) y = sin
–1
 
?
?
?
?
?
?
?
?
?
?
2 x
1 x
2
2
Sol. (a) We have y = 
x 3 cos x 3 sin 2
1
? ?
  i.e. sin 3x + cos 3x = 
y
1
 – 2
i.e.
2
 sin 
1
3x
4 y
? ? ?
? ?
? ?
? ?
 – 2 i.e. sin 
1
3x
4 2
? ? ?
? ?
? ?
? ?
?
?
?
?
?
?
?
?
? 2
y
1
since, 
?
?
?
?
?
? ?
?
4
x 3 sin
 ? 1, therefore we have
2
y
1
?
 ? 
2
i.e. –
y
1
2 ?
 – 2 ? 
2
i.e. 2 – 
y
1
2 ? ? 2 2
i.e.
2 2
1
y
2 2
1
?
? ?
?
Hence, the range is y ? ?
?
?
?
?
?
? ? 2 2
1
,
2 2
1
.
(b) We have
2 x
1
1
2 x
1 x
2 2
2
?
? ?
?
?
Now, we have2 ? x
2
 + 2 < ? i.e.
2 x
1
2
1
2
?
?
 > 0 i.e.  
2 x
1
2
1
2
?
?
?
?
 < 0
i.e. 1 – 
2 x
1
1
2
1
2
?
? ?
 < 1 i.e.
2 x
1 x
2
1
2
2
?
?
?
 < 1 i.e.  sin
–1
 
2
1
 ? sin
–1
 
?
?
?
?
?
?
?
?
?
?
2 x
1 x
2
2
 < sin
–1
 1
gives
2
y
6
?
? ?
?
Hence, the range is y ? 
?
?
?
?
?
? ? ?
2
,
6
.
Ex.7 Find the range of following functions :  (i) y = ln (2x – x
2
)    (ii) y = sec
–1
 (x
2
 + 3x + 1)
Sol. (i) using maxima–minima, we have (2x – x
2
) ? (– ?, 1]
For log to be defined accepted values are 2x – x
2
 ? (0, 1] {i.e. domain (0, 1]}
ln (2x – x
2
) ? (0, 1] ? range is (– ?, 0]
(ii) y = sec
–1
 (x
2
 + 3x + 1)
Let t = x
2
 + 3x + 1 for x ? R then t ? 
?
?
?
?
?
?
? ? ,
4
5
but y = sec
–1
 (t) ? ?? ? ?t ? ?
?
?
?
?
?
? ? 1 ,
4
5
 ? [1, ?)
from graph range is y ? ?
?
?
?
?
?
? ?
?
?
?
?
?
? ? ?
?
?
?
?
? ?
?
,
4
5
sec
2
, 0
1
       
–1
–5/4 0 1
t
y
?/2
?
sec (–5/4)
–1
Ex.8 Find the range of y = 
?
?
?
?
?
?
? ?
?
) 1 x x ln(sin
2 1
Sol. We have x
2
 + x + 1 = 
2
2
1
x ?
?
?
?
?
?
?
 + 
4
3
 which is a positive quantity whose minimum value is 3/4.
Also, for the function y = 
?
?
?
?
?
?
? ?
?
) 1 x x ln(sin
2 1
 to be defined, we have x
2
 + x+ 1 ? 1
Thus, we have  
4
3
 ? x
2
 + x + 1 ? 1   i.e.   ? ? ? ? 1 x x
2
3
2
    i.e.  
2
) 1 x x ( sin
3
2 1
?
? ? ? ?
?
?
[ ? sin
–1
 x is an increasing function, the inequality sign remains same]
i.e. ln
?
?
?
?
?
? ?
3
 ? ln(sin
–1
1 x x
2
? ?
 ? ln 
?
?
?
?
?
? ?
2
i.e. 0.046 ? ln(sin
–1
 
1 x x
2
? ?
)] = 0. Hence, the range is y ? ?[ ?n ?/3, ?n ?/2]
Page 5


 
 
C. IMPORTANT TYPE OF FUNCTIONS
(1) Trigonometric function :
Function Domain Range         Curve
     (i) f(x) = sin x          x ? R y ? [–1, 1]   x
y = sin x
1
–1
2 / 3 ?
2 / ? ? 2
? ?2 ? ? 2 / 3 ? ? ?
2 / ? ?
     (ii) f(x) = cos x x ? R y ? [–1, 1]     
x
y = cos x
1
–1
2 / 3 ?
2 / ? ? 2
? ?2
? ?
2 / 3 ? ?
?
2 / ? ?
2 / 5 ?
     (iii) f(x) = tan x x ? R – (2n + 1) 
2
?
, n ? ? ? y ? R       
y = tan x
x
O
2
? ?
2
3 ?
2
3 ?
? ? ?
2
?
?
     (iv) f(x) = cot x x ? R – n ? ? n ? ? ? y ? R
y = cot x
x
O
2
?
?
2
3 ?
? ?
2
?
?
? 2
     (v) f(x) = cosec x x ? R –n ?, n ? ? ? ? ? ??? ? y ? ?? ? ?, –1] ? ?[1, ?)     
1
y = cosec x
2 / ?
0
–1
2 / ? ?
? ?
2 / 3 ?
? 2
x
?
     (vi) f(x) = sec x   x ?R – (2n+1) 
2
?
, n ? ?   y ? ?? ? ?, –1] ? ?[1, ?)  
1
y = sec x
0
–1
2 / ? ? 2 / 3 ?
x
?
2 / ?
? ?
2 / 3 ? ?
(2) Polynomial Function :
f(x) = a
0
x
n
 + a
1
x
n – 1
 + a
2
 x
n–2
 +.......+ a
n
where a
0
, a
1
, a
2
 ........ a
n
 ? R n ? W
If a
0
 ? 0, then f(x) is called n
th
 degree polynomial and Domain x ? R
(3) Algebraic Function : A function is called an algebraic function. If it can be constructed using
algebraic operations such as additions, subtractions, multiplication, division taking roots etc.
All polynomial functions are algebraic but converse is not true.
Ex : f(x) = 
2 4
x 5 x ?
 + x + (x
3
 + 5)
3/5
, f(x) = x
3
 + 3x
2
 + x + 5
Remark : Function which are not algebraic are called as TRANSCENDENTAL FUNCTION.
Ex : f(x) = 
3
5 / 3 2 5
x
) x 5 x ( ?
 + 3 6 x 5 x
2
? ? + ?n x  ? transcidental function
Ex : f(x) = 7 x
2
? + e
?n x
 + 
7 x
7 x
2
?
?
  ? algebraic function.
(4) Rational Function : It is a function of form f(x) = 
) x ( h
) x ( g
, where g(x) & h(x) are poly. function and
h(x) ? 0 Ex. f(x) = 
4 x
2 x 3 x
2
2 4
?
? ?
(5) Logarithmic function : f(x) = log
a
x, where x > 0, a > 0, a ? 1
a ? base, x ? number or argument of log.
Case–I : 0 < a < 1 Case–II : a > 1
             f(x) = log
a
 x    f(x) = ?n x
             Domain : x ? (0, ?)     
O
(1, 0)
x
f(x)
   
O (1, 0) x
f(x)
             Range : y ? R
(6) Exponential function : f(x) = a
x
, where a > 0, a ? 1
a ? Base x ? Exponent
Case–I : 0 < a < 1 ; a = 1/2 Case–II : a > 1
             f(x) = 
x
1
2
? ?
? ?
? ?
    
O
(1, 0)
x
f(x)
O
(0,1)
x
f(x)
             Domain : x ? R
             Range : y ? (0, ?)
(7) Absolute value function (Modulus function) :
y = |x| = 
?
?
?
? ?
?
0 x ; x
0 x ; x
y   x = y=–x
y
x
Domain : x ? R; Range : y ? R
+
 ? {0}
(8) Signum function :
y = sgn (x) = 
?
?
?
?
?
? ?
?
?
0 x ; 1
0 x ; 0
0 x ; 1
   
y
1
0
–1
x
Domain : x ? R; Range : y ? {–1, 0, 1}
(9) Greatest integer function (step-up function) :
y = f(x) = [x] 
?
?
?
?
? ?
x than less
otherwise ; Integer Greatest
I x ; x
1 2
–1
0
1
2
y
3 x
Domain : x ? R; Range : y ? I
Ex : [2
.
3] = 2, [5] = 5, [–2
.
3] = –3
Properties :
(i) [x] ? x < [x] + 1 (ii) [x + m] = [x] + m ; m ? I (iii) [x] + [–x] = 
?
?
?
? ?
?
I
I
x ; 1
x ; 0
(10) Fractional part function :
y = f(x) = {x} = x – [x]
Domain : x ? R; Range : [0, 1)
Ex : 2.3 = 2 + 0.3 ? ?fractional part
1 –1 2 –2 0
y
3 x
1
               ?
        Integer part
Properties :
(i) Fractional part of any integer is zero. (ii) {x + n} = {x}, n ? ? ?
(iii) {x} + {–x} = 
?
I 0 ; x
1 ; otherwise
?
Ex.6 Find the range of the following functions : (a) y = 
x 3 cos x 3 sin 2
1
? ?
  (b) y = sin
–1
 
?
?
?
?
?
?
?
?
?
?
2 x
1 x
2
2
Sol. (a) We have y = 
x 3 cos x 3 sin 2
1
? ?
  i.e. sin 3x + cos 3x = 
y
1
 – 2
i.e.
2
 sin 
1
3x
4 y
? ? ?
? ?
? ?
? ?
 – 2 i.e. sin 
1
3x
4 2
? ? ?
? ?
? ?
? ?
?
?
?
?
?
?
?
?
? 2
y
1
since, 
?
?
?
?
?
? ?
?
4
x 3 sin
 ? 1, therefore we have
2
y
1
?
 ? 
2
i.e. –
y
1
2 ?
 – 2 ? 
2
i.e. 2 – 
y
1
2 ? ? 2 2
i.e.
2 2
1
y
2 2
1
?
? ?
?
Hence, the range is y ? ?
?
?
?
?
?
? ? 2 2
1
,
2 2
1
.
(b) We have
2 x
1
1
2 x
1 x
2 2
2
?
? ?
?
?
Now, we have2 ? x
2
 + 2 < ? i.e.
2 x
1
2
1
2
?
?
 > 0 i.e.  
2 x
1
2
1
2
?
?
?
?
 < 0
i.e. 1 – 
2 x
1
1
2
1
2
?
? ?
 < 1 i.e.
2 x
1 x
2
1
2
2
?
?
?
 < 1 i.e.  sin
–1
 
2
1
 ? sin
–1
 
?
?
?
?
?
?
?
?
?
?
2 x
1 x
2
2
 < sin
–1
 1
gives
2
y
6
?
? ?
?
Hence, the range is y ? 
?
?
?
?
?
? ? ?
2
,
6
.
Ex.7 Find the range of following functions :  (i) y = ln (2x – x
2
)    (ii) y = sec
–1
 (x
2
 + 3x + 1)
Sol. (i) using maxima–minima, we have (2x – x
2
) ? (– ?, 1]
For log to be defined accepted values are 2x – x
2
 ? (0, 1] {i.e. domain (0, 1]}
ln (2x – x
2
) ? (0, 1] ? range is (– ?, 0]
(ii) y = sec
–1
 (x
2
 + 3x + 1)
Let t = x
2
 + 3x + 1 for x ? R then t ? 
?
?
?
?
?
?
? ? ,
4
5
but y = sec
–1
 (t) ? ?? ? ?t ? ?
?
?
?
?
?
? ? 1 ,
4
5
 ? [1, ?)
from graph range is y ? ?
?
?
?
?
?
? ?
?
?
?
?
?
? ? ?
?
?
?
?
? ?
?
,
4
5
sec
2
, 0
1
       
–1
–5/4 0 1
t
y
?/2
?
sec (–5/4)
–1
Ex.8 Find the range of y = 
?
?
?
?
?
?
? ?
?
) 1 x x ln(sin
2 1
Sol. We have x
2
 + x + 1 = 
2
2
1
x ?
?
?
?
?
?
?
 + 
4
3
 which is a positive quantity whose minimum value is 3/4.
Also, for the function y = 
?
?
?
?
?
?
? ?
?
) 1 x x ln(sin
2 1
 to be defined, we have x
2
 + x+ 1 ? 1
Thus, we have  
4
3
 ? x
2
 + x + 1 ? 1   i.e.   ? ? ? ? 1 x x
2
3
2
    i.e.  
2
) 1 x x ( sin
3
2 1
?
? ? ? ?
?
?
[ ? sin
–1
 x is an increasing function, the inequality sign remains same]
i.e. ln
?
?
?
?
?
? ?
3
 ? ln(sin
–1
1 x x
2
? ?
 ? ln 
?
?
?
?
?
? ?
2
i.e. 0.046 ? ln(sin
–1
 
1 x x
2
? ?
)] = 0. Hence, the range is y ? ?[ ?n ?/3, ?n ?/2]
Ex.9 f : R ? R,  f (x) =
1 x
n m x x 3
2
2
?
? ?
. If the range of this function is [– 4, 3) then find the value of (m
2
 + n
2
).
Sol. f (x) = 
2
2
x 1
3 n m x ) 1 x ( 3
?
? ? ? ?
; f (x) = 3 + 
2
x 1
3 n m x
?
? ?
y = 3 + 
2
x 1
3 n m x
?
? ?
for  y to lie in [– 4, 3) mx + n – 3 < 0   ?  x ? R
this is possible only if  m = 0 when,  m = 0    then  y = 3 + 
2
x 1
3 n
?
?
note that  n – 3 < 0  (think !) n < 3 if   x ? ?,   y
max
 ? 3
–
now y
min
 occurs at x = 0 (as 1 + x
2
 is minimum)
y
min
 = 3 + n – 3 = n ?  n = – 4 so m
2
 + n
2
 = 16
Ex.10 Find the domain and range of f(x) = sin 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
x 1
x 4
n
2
?
Sol.
2
x 4 ?
 is positive and x
2
 < 4 ? –2 < x < 2
1 – x should also be positive. ? x < 1
Thus the domain of 
?
?
?
?
?
?
?
?
?
?
x 1
x 4
n
2
? is –2 < x < 1 sine being defined for all values, the domain of sin
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
x 1
x 4
n
2
?
 is the same as the domain of 
?
?
?
?
?
?
?
?
?
?
x 1
x 4
n
2
?
To study the range. Consider the function 
x 1
x 4
2
?
?
As x varies from –2 to 1, 
x 1
x 4
2
?
?
 varies in the open interval (0, ?) and hence 
x 1
x 4
n
2
?
?
? varies from
– ? to + ?. Therefore the range of sin 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
x 1
x 4
n
2
?
 is (–1, +1)
Ex.11 Find the range of the function f(x) = sin
–1
 
10
4
x 5 1
x 1
?
?
.
Sol. Consider g(x) = 
10
4
x 5 1
x 1
?
?
. Also g(x) is positive ? x ? R and g(x) is continuous ? x ? R and
g(0) = 1 and 
? ? x
lim
g(x) = 0
? g(x) can take all values from (0, 1] ? Range of f(x) = sin
–1
 (g(x)) is ?
?
?
?
?
? ?
2
, 0
.
Ex.12 f(x) = cos
–1
 {log [ ] 1 x [
3
? ]}, find the domain and range of f(x ? (where [ 
*
 ] denotes the greatest in-
teger function).
Sol. If cos
–1
 x = ?, then– 1 ? x ? 1 ? –1 ? log ] ] 1 x [ [
3
? ? 1 ? e
–1
 ? ] ] 1 x [ [
3
? ? e
0.37 ? ? ] ] 1 x [ [
3
? ? ?2.7 ? 1 ? ] 1 x [
3
? < 3 ? 1 ? [x
3
 + 1] < 9
1 ? [x
3
] + 1 < 9 0 ? [x
3
] < 8 ? 0 ? ?x < 2
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