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2. Relations and Functions
QUESTION 1:Identify the graphs given .Write their name and describe the function.
(i)
(ii)
(iii)
SOLUTION:
(i)Identity Function y=x
(ii)The greatest integer function Function
y = [ x]
(iii)Signum Function:y = sgn(x)
QUESTION 2: Tabulated below are some Standard Real Functions with their
Domain and Range. Fill in the blanks in the table.
Function
Constant
function
Identity
function
Greatest
Integer
Defining Rule
f(x) = k
Domain
R
Range
k
f(x) = x
R
R
f(x) = [x]
R
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function
Modulus
function
Signum
function
f(x) =||x||
 1, x > 0

f(x) =  0, x = 0
 −1, x < 0

SOLUTION:
Function
Constant
function
Identity
function
Greatest
Integer
function
Modulus
R
R+ ∪ {0}
Positive
Square
root
function
Exponent
ial
function
Logarithmic
function
R+ ∪ {0}
R+ ∪ {0}
f(x) = a x
( a >0 , a≠
≠1)
R
f(x) =loga x
( a >0 , a≠
≠1)
R+
Defining Rule
f(x) = k
Domain
R
Range
k
f(x) = x
R
R
f(x) = [x]
R
Z
f(x) =||x||
R
R+ ∪ {0}
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function
Signum
function
Positive
Square
root
function
Exponent
ial
function
Logarithmic
function
QUESTION 3:
 1, x > 0

f(x) =  0, x = 0
 −1, x < 0

R
{-1,0,1}
f(x) =√x
R+ ∪ {0}
R+ ∪ {0}
f(x) = a x
( a >0 , a≠
≠1)
R
R+
f(x) =loga x
( a >0 , a≠
≠1)
R+
R
x2; 0 ≤ x ≤ 4
f(x) = 
,and
4
x
;
4
≤
x
≤
15

x2; 0 ≤ x ≤ 5
g(x)= 
4 x; 5 ≤ x ≤ 15
f(x) and g(x) are two relations .(i) Is f(x) a function?(ii) Is g(x) a function?
SOLUTION:
(i) In the relation f, for ∀x ∈ [0,15], there is a unique value in f(x)
∴The relation f ( x)is a function .
(ii ) In the relation g, for ∀x ∈ {[0,5) ∪ (5,15], there is a unique value in g(x)
But for x = 5, g ( x) = x 2 = 25and g(x) = 4 x = 20
i.e x = 5 ⇒ g ( x) = 20, 25
There is more than one value of g(x) for x = 5
∴The relationg ( x)is not a function
QUESTION 4:
If f ( x ) = x 2 , then find
f(4.4) - f(4)
4.4 - 4
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SOLUTION:
f ( x) = x 2
f (1) = 2 2 = 4; f (2.2) = ( 2.2 ) = 4.84
2
f(4.4) - f(4) 19.36 − 16 3.36
=
=
= 8.4
4.4 - 4
0.4
0.4
QUESTION 5: Find the domain and range of the real function f(x) = x+3
∴
SOLUTION:
f ( x) = x + 3
A unique value of f(x) exists forevery real value of x
∴ Domain of f =R
 x, x ≥ 0
We know, x = 
 − x, x < 0
 ( x + 3) , x ≥ −3
∴ x-1 = 
− ( x + 3) , x < −3
⇒ x + 3 takes non-negative real values
Range of f =[0,∞)
QUESTION 6:
Let f = {(1,1) (2,3),...} be a function from Z to Z, defined by f(x) = ax +b , for some integers a and b.
Determine a and b.
SOLUTION:
f ( x) = ax + b
(1,1) ∈ f ⇒ f (1) = a.1 + b = 1 ⇒ a + b = 1
(2,3) ∈ f ⇒ f (2) = a.2 + b = 3 ⇒ 2a + b = 3
solving the two equations , we get a = 2, b=-1
QUESTION 7:
If f and g are two functions : R → R;f ( x) = 2 x − 1, g ( x) = 2 x + 3, Then evaluate
(i ) ( f + g ) ( x)
(ii ) ( f − g ) ( x)
f 
(iii) ( fg ) ( x) (iv)   ( x)
g
SOLUTION:
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f ( x) = 2 x − 1, g ( x) = 2 x + 3,; x ∈ R
(f
(f
+ g ) ( x) = ( 2 x − 1) + ( 2 x + 3 ) = 4 x + 2; x ∈ R
− g ) ( x) = ( 2 x − 1) − ( 2 x + 3 ) = − x − 4 = −(4 + x); x ∈ R
( fg )( x) = ( 2 x − 1)( 2 x + 3) = 4 x 2 − 2 x + 6 x − 3 = 4 x 2 + 4 x − 3
 f 
2x −1
 3
; x ∈ R − − 
  ( x) =
2x + 3
 2
g
QUESTION 8:
Let A = {a,b,c,d} ; B = {a, e, i, k , l , m} and f = {( a,a ) , ( a, e ) , ( b, i ) , ( d , p )}
Is f a relation from A to B?
SOLUTION:
f = {( a,a ) , ( a, e ) , ( b, i ) , ( d , p )}
( a,a ) , ( a, e ) , ( b, i ) ∈ A × B
(d, p) ∉ A× B
⇒ f is not a relation
QUESTION 9:
Find the domain and range of f(x) = √[x-7]
SOLUTION:
Let y = f ( x) = x − 7
Domain of f : x − 7 ≥ 0[Square root is defined only for non-negative reals]
⇒x≥7
⇒ Domain of f=[7,∞)
Now, x − 7 = y
⇒ ( x − 7) = y 2
⇒ x = y2 + 7
Range of f: y ≥ 0[Square root is for non-negative for all reals, since we are only dealing with real functions ]
Range of f = [0,∞)
QUESTION 10:
Find the domain and range of f(x) = 3 / 5 - x 2
SOLUTION:f(x) = 3 / 5 - x 2
For domain: 5 - x2 should not equal zero, i.e 5 shold not equal x 2
Domain = { x : x R , x sq root 5
For range
5y - x2y=3
x2= (5y-3)/y
x = Sq root (5y-3)/y
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For (5y-3)/y to be positive either both Nr and Dr should be positive , in which case y>3/5
. Or else both should be negative , in which case y <0.
Range is therefore all real numbers except between 0 and 3/5 ={x:x∈R,x≠0,3/5}
QUESTION 11:
x 2 − 11
Find the domain of the function f(x) = 2
x − 8 x + 12
SOLUTION:
Forf ( x) to be defined : x 2 − 8 x + 12 ≠
⇒ ( x − 2)( x − 6) ≠ 0
⇒ x ≠ 2; x ≠ 6
Domain = R − {2, 6}
QUESTION 12:
.From the given table , is y a function of x . Justify your answer .
SOLUTION:The relation in the above table is function as no value of x is
repeated in it.
QUESTION 13:
3 − x, x > 1

Draw the graph of f(x)  1, x = 1 and find the Range of f .
 2 x, x < 1

SOLUTION:
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QUESTION14: If : R → R; f ( x) =
x2
.What is the range of f?
x2 + 1
SOLUTION:
x2
x2 + 1
Domain of f(x) = R
f ( x) =
x2
x2 + 1
x 2 ≥ 0 ⇒ x 2 + 1 ≥ 1 ⇒ Denominator ≥ Numerator ⇒ y < 1
Range of f(x) : Let y = f ( x) =
x2
Now, y = 2
⇒ y ( x 2 + 1) = x 2 ⇒ yx 2 + y = x 2 ⇒ x 2 ( y − 1) = − y
x +1
y
⇒ x2 =
1− y
⇒ 1− y ≠ 0
⇒ y ≠1
y
≥0
1− y
⇒ Case1: y ≥ 0;1 − y ≥ 0 ⇒ y ≥ 0;1 ≥ y or y ≤ 1,buty ≠ 1
Now, x 2 =
i.e y ∈ [0,1)
Case2 : y ≤ 0;1 − y ≤ 0 ⇒ y ≤ 0;1 ≤ y or y ≥ 1
Not possible for both conditions to be satisfied simultaneously
∴ Range of f(x) = [0,1)
QUESTION15: Let A = {a, b} and B = {c, d}. Find the number of relations from A to
B.
SOLUTION: We have, A x B = {(a, c), (a, d), (b, c), (b, d)}.
Since n (A x B ) = 4, the number of subsets of A⋅B is 24. Therefore, the number of
relations from A into B will be 24.
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2. RELATION AND FUNCTIONS
1. Let A = {1,2,3} ,B = {4} and C = {5} . Verify that
(i)
( ii )
A × (B ∪ C) = ( A × B) ∪ ( A × C)
A × (B − C) = ( A × B) − ( A × C) .
2. Let A = {1, 2} ,B = {1, 2, 3, 4} ,C = {5, 6} and D = {5, 6,7, 8} . Verify that
A × C ⊂ B × D.
3. Determine the domain and the range of the relation R defined by
R = {( x + 1, x + 5 ) : x ∈ {0,1,2, 3, 4,5}} .
4. Determine the domain and the range of the following relations
( i ) {(1, 2) , (1,6 ) , (1,8 )}
( ii ) {( x, y ) : x ∈ N, y ∈ N and x + y = 10}
( iii ) {( x, y ) : x ∈ N, x < 5,
( iv ) {( x, y ) : y =
y = 3}
x − 1 , x ∈ Z and x ≤ 3}
5. Find the domain and range of the following functions:
(i)
 x 2 − 1 

 : x ∈ R, x ≠ 1 .
 x,
x −1 


{( x, 9 - x ) : x ∈ R}.
( iii )
2
( ii ) {( x, − x ) : x ∈ R} .
( iv )

1
 x,
2
 1 − x


 : x ∈ R, x ≠  .


6. Draw the graph of the following functions:
( i ) f : R → R such that f ( x ) = 4 − 2x.
( ii ) f : R → R such that f ( x ) = x − 2 .
7. Let f : Z → Z, g : Z → Z be functions defined by f =
g=
{( n, n ) : n ∈ Z} and
2
{( n, n ) : n ∈ Z}. Show that f = g.
2
8. Let A = {1,2, 3, 4} ,B = {1,5,9,11,15,16} and f = {(1,5 ) , ( 2, 9 ) , ( 3,1) ,(4,5), ( 2,11)} .
Are the following true?
( i ) f is a relation from A into B
( ii )
f is a function from A into B?
Justify your answer in each case.
9. Let A ⊆ N and f : A → A be defined by f ( n ) = p, the highest prime factor of n such
that the range of f is A. Determine A.
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Answers
3. Domain = {1, 2, 3, 4,5, 6} , Range = ( 5,6, 7,8, 9,10 ) .
4.
( i ) Domain = {1} ,Range = {2, 4,6, 8}
( ii ) Domain = {1, 2, 3, 4,5, 6,7,8,9} , Range = {9,8,7,6,5, 4, 3, 2,1} .
( iii ) Domain = {1,2,3, 4} , Range = {3}
( iv ) Domain = {−3, −2, −1,0,1,2,3} , Range = {4, 3,1,0,2}
5.
(i)
( ii )
Domain = R − {1} , Range = R − {2}
( iii )
Domain = {x : x ∈ Rand − 3 ≤ x ≤ 3} Range = { y : y ∈ R and − 3 ≤ y ≤ 3}
Domain = R, Range = { y : y ∈ R and y ≤ 0}
( iv ) Domain = R − {1, −1} ,Range = { y : y ∈ R,
8.
(i)
Yes
( ii )
y ≠ 0, y < 0 and y ≥ 1}
No
9. A set of prime numbers
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