FUNCTION PROBLEMS FOR JEE ANS STATE BOARD PRACTICE

1. FUNCTIONS
FUNCTIONS

2. FUNCTIONS
PROBLEMS BASED ON
DOMAIN & RANGE

3. FUNCTIONS
1. Find the domain of the real function f(x) =
1
(x2 − 1)(x + 3)
f(x) is not defined when,
(x2-1)(x+3)=0
(x-1)(x+1)(x+3)=0
x =1,-1,-3
domain of f is R-{1, -1, -3}
Solution
(x-1)=0 or (x+1)=0 or (x+3)=0
Given function is
in the form of
1
f(x)
Here, what is
the domain of
f(x)?

4. FUNCTIONS
2. Find the domain of the real function
2x2 − 5x + 7
(x − 1) (x − 2) (x − 3)
Here f(x) is not defined when,
(x-1)(x-2)(x-3)=0
Let f(x) =
2x2 − 5x + 7
(x−1)(x−2)(x−3)
Solution
(x-1)=0 or (x -2)=0 or (x-3) = 0
Where f(x) is not
defined?
 x = 1,2,3
domain of f is R-{1, 2, 3}
Guess the
domain of f(x).

5. FUNCTIONS
3. Find the domain of the real function f(x)= 16 − x𝟐
Domain of f(x) is [-4, 4]
16−x𝟐 is defined when,
x2 -16  0
x  [-4, 4]
Solution
(x + 4) (x - 4)  0
16 - x2  0
When f(x) is
defined?
If
(x-) (x-)≤0x[,]
(x-(-4)(x - 4)  0

6. FUNCTIONS
Domain of f(x) is (-, -5][5, )
4. Find the domain of the real function f(x) = x𝟐 − 25
x𝟐 − 25 is defined when,
(x-5)(x+5)  0
x(-, -5][5, )
Solution
x𝟐
− 25  0
x-5 x  5
(or)
When f(x) is
defined?
(x-5)(x-(-5))  0
If
(x-) (x-)≥0x(-, ][, )
x≤ or x≥ 

7. FUNCTIONS
5. Find the domain of the real function f(x) = 4x − x𝟐
Domain of f(x) is [0,4]
4x − x𝟐 is defined when,
x𝟐
-4x  0
x[0,4]
(x-0)(x-4)  0
Solution
4x − x𝟐
 0
x(x-4)  0
0x 4
When f(x) is
defined?
If
(x-) (x-)≤0x[,]

8. FUNCTIONS
6. Find the domain of the real function f(x) = 2−x + 1+x
x  -1(2)
2−x is defined when,
x - 2  0
1+x is defined when,
Solution
2 - x  0
x  2(1)
1 + x  0
Here we find
domain of
2−x , 1+x
seperately
When 2−x is
defined?
When 1+x is
defined?
-2 -1 0 1 2 
-
Hence from (1)
-1  x  2 x  [-1, 2]
and (2)

9. FUNCTIONS
7. Find the domain of the real function f(x) =
1
1 − x𝟐
f(x) is defined when,
x2 – 1 < 0
Solution
(x +1) (x -1) < 0
1 - x2 > 0
(x –(-1)) (x -1) < 0
Domain of f(x) is (-1,1)
x(-1, 1)
-1 < x < 1
If
(x-) (x-)<0x(,)

10. FUNCTIONS
8. Find the domain of the real function log(x2 - 4x + 3)
Let f(x)=log(x2 -4x + 3)
f(x) is defined when
(x - 1) (x - 3) > 0
Domain of f(x) is (-, 1)(3, )
x (-, 1)(3, )
Solution
x2 - 4x + 3 > 0
If (x-) (x-)>0x(-,)(,)
x< or x>

11. FUNCTIONS
9. Find the domain of the real function f(x) =
1
log(2 − x)
2 - x > 0 and 2 - x  1
x - 2 < 0 and 2 -1  x
f(x) =
1
log(2 − x)
is defined when
Solution :
x < 2 and x  1
2
1
0
-1
-2
×
-∞ +∞
(OR) x(-∞,1)(1,2)
x (-, 2)-{1}

12. FUNCTIONS
10. Find the domain of the real function
2 + x + 2 − x
x
Let f(x) =
2 + x + 2 − x
x
2 + x is defined when
x + 2  0
2−x is defined when
Solution
2 + x  0
x  -2  (1)
2 - x  0
x - 2  0
x  2  (2)

13. FUNCTIONS
2
1
0
-1
-2
×
-∞ +∞
From (1), (2) and (3)
1
x
is defined when x  0(3)
x  -2  (1)
x  2  (2)
 x  [-2, 0)  (0, 2]

14. FUNCTIONS
11.Find the domain of the real valued function
f(x)= log0.3(x − x𝟐)
log0.3(x−x𝟐) is defined when
Solution 0 < (x - x2)  1
(x - x2) >0 and (x - x2)  1
i) x - x2 > 0
x2 - x < 0
x (x - 1) < 0
(x-0)(x-1)<0
 x  (0, 1)

15. FUNCTIONS
Also ii) (x - x2)  1
x2 – x +1  0
x −
1
2
𝟐
+
3
4
 0 (True  x R)
 domain = (0, 1)  R = (0, 1)

16. FUNCTIONS
12. Find the domain f(x)= x − [x]
Domain of f(x) is R
x − [x] is defined when
Solution
x - [x]  0 
which is true  x  R
x  [x]
When f(x)
is defined?

17. FUNCTIONS
13. Find the domain of f(x)= [x]−x
But [x] > x is not possible.
f(x) exists when
[x] > x (or)
Now [x] = x
Domain of f(x) is Z
Solution
[x] - x  0
[x]  x
[x] = x
 x  Z
So, we must have [x] = x
Is it
possible?

18. FUNCTIONS
14. Find the domain f(x) = x − x
Domain of f(x) is R
x − x is defined when
Solution
x − x  0 which is true  x  R

19. FUNCTIONS
15. Find the domain of f(x) = log(x - [x]) ?
 Domain of f(x) is R - Z.
log(x - [x]) is defined when
Solution
x - [x] > 0
x > [x] which is true  x R - Z.

20. FUNCTIONS
16. Find the domain and range of f(x) =
x
2 − 3x
f(x) =
x
2 − 3x
does not exists when,
 Domain of f is R -
2
3
Let y = f(x)
Solution
2 - 3x = 0
x =
2
3
 y =
x
2 − 3x
When is
f(x) not
defined?

21. FUNCTIONS
2y - 3xy = x
x =
2y
1 + 3y
Range of f is R -
−1
3
y (2 - 3x) = x
2y = x (1 + 3y)
which is defined when, 1 + 3y  0
y 
−1
3
2y = x + 3xy

22. FUNCTIONS
17. Find the domain and range of the f(x)= 9−x𝟐
Domain of f(x) is [-3, 3]
(i) Domain
f(x)= 9 − x𝟐 is defined when,
x2 – 9  0
x  [-3, 3]
Solution
9 - x2  0
(x + 3) (x - 3)  0
-3  x  3
(x – (-3)) (x - 3)  0

23. FUNCTIONS
(ii) Range
We know that x2  0
The range of the f(x) is [0, 3]
-x2  0
9 – x2  9
9 − x2  3
and We know that 9−x2  0
𝟎 9−x2 (2)
From (1) & (2), 𝟎  9 − x2  3
f(x)= 9−x𝟐 0,  x  [-3,3]
(1)

24. FUNCTIONS
18. Find the range of the real function
x𝟐
− 4
x − 2
Let f ( x ) =
x𝟐
− 4
x − 2
Solution
f ( x ) does not exist when x – 2 = 0
x = 2
Domain = R - { 2 }
Also f(x) =
x𝟐
− 4
x − 2
= x + 2
=
(x + 2) (x − 2)
(x − 2)
When is
f(x) not
defined?

25. FUNCTIONS
Now f ( x ) = x + 2
Thus f ( x ) can take any value except 4.
Range = R - { 4 }.
But x  2
x + 2  2 + 2
f ( x )  4

26. FUNCTIONS
19. Find the domain and range of the real valued function
f(x)=
2+x
2−x
Domain is R-{2}
Let f ( x ) =
2 + x
2 − x
Let y =
2 + x
2 − x
Solution
2 – x = 0
f ( x ) does not exist when,
x = 2
y ( 2 – x ) = 2 + x

27. FUNCTIONS
y +1  0
Range is R - {-1}
2 y – x y = 2 + x
 x =
2 y − 2
y + 1
2 y – 2 = x + x y
2 y - 2 = x ( 1 +y )
which is defined when,
y  -1
2 y − 2
y + 1
= x

28. FUNCTIONS
20.Find the domain and range of the function
f(x)=|x|+|1+x|
f(x) =|x|+|1 + x| is defined
Domain is R.
We know u + v  u + v for u, v  R
Put u = -x , v = 1 + x in above equation
x + 𝟏 + x  1
f(x)  1
Solution
1  −x + 𝟏 + x
−x + 1 + x  −x + 1 + x
( ∵ −x = x )
Range = [1, ∞)

29. FUNCTIONS
Thank you…