(a) The Cartesian product $P \times P$ has 9 elements among which are found (-2, 0) and (0, 2). the set P is _____ and the remaining elements of P × P__________
(b) The function f(x) is defined as $x^2$ . The value of $ \frac {f(2) -f(1)}{2-1}$ ______
(c) The function p(x)=x+1 and q(x)=2x-1.The value (f/g)x is ______
(d) The Function g(x)=6x2+3x-2 The value of g(-1) is ________
a. Since n(P × P) =9, Set P has three elements. Now since (–2, 0) and (0, 2) belongs P × P, Set P ={-2,0,2}
Remaining elements in P × P are
{(-2,-2),(-2,2),(0,-2),(0,0),(2,-2),(2,0),(2,2)}
b. 3
c. $\frac {x+1}{2x -1}$
d. $g(-1) = 6 (-1)^2 + 3 (-1) -2 =1$
True or False statement
(1) The relation defined as {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)} is a function
(2) The relation defined as {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)} is not a function
(3) The relation defined as {(1, 3), (1, 5), (2, 5)} is a function
(4) The below graph is not a function
(5) A function are relations but all relations are not functions
(6) A function is defined as
$f(x) =- \sqrt {-2x+5}$
The domain is x≤ 5/2 and Range is f(x) ≤ 0
(7) P= {1,2,3} Q={e,f}. The total number of relation from P × Q is 64
(8) The below graph is a function
(9) The below graph is not a function
(10) The ordered pair {(x,y)|y < 3x+1} is a function
(11) The ordered pair {(x,y)|y=x2} is a relation but not function
(12) The ordered pair {(x,y)| x=3 and y is real number} is a relation and function Solution
T
F
F
F
T
T
T
T
T
F
F
F
Subjective Questions
Find the domain and range of the following real function:
(1) $y=x^2$
(2) $y=-|x|$
(3) $y=3x-7$
(4) $y=-x^4 + 3$
(5) $y= \sqrt {2-x}$
(6) $y =\frac {1}{ \sqrt {11-x}}$ Solution
1. $y=x^2$
Domain is all the Real number as function is defined for all values
Domain =R
The function always provides positive value. So range is $[0,\infty)$
2.$y=-|x|$
Domain is all the Real number as function is defined for all values
Domain =R
The function always provides negative value. So range is $(-\infty,0]$
3.$y=3x-7$
Domain is all the Real number as function is defined for all values
Domain =R
The is a linear function. Range is also R
4.$y=-x^4 + 3$
Domain is all the Real number as function is defined for all values
Domain =R
Now it can be written as
$y=3 -x^4$
Now $x^4$ will always be positive for all real values of x,So Range will be $(-\infty,3]$
5. $y= \sqrt {2-x}$
Now this function is defined for x where $2 -x \geq 0$
or $ x \leq 2 $
So Domain is $ (-\infty,2] $
Since square root gives positive values only, Range is $[0,\infty)$
6.$y =\frac {1}{ \sqrt {11-x}}$
Now this function is defined for x where $11 -x > 0$ as
or $ x < 11$
So Domain is $(-\infty,11)$
Since square root gives positive values only.Also this function cannot have zero value, So Range is $(0,\infty)$
Linked Type comprehension
If A= {1,2,3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} D = { 17,19,21,23} and E={1,-1} find
(i) What is the number of element in $A \times B$
(ii) How many number of relations can be found from $A \times C$
(iii) The mapping defined as {(1,11),{1,13},{2,7},{5,11} is a function from A × B,State True or False
(iv) The mapping defined as {(11,17),(13,19),(15,21),(15,23)} is a Relation from C × D, State True or False
(v) Find the value of B × C and C × B
(vi) Find the value of E × E × E
(vii) Verify that A × (B ∩ C) = (A × B) ∩ (A × C)
(viii) A X B is a subset of A X C. State True or false
(ix) (A X B) ∩ (B ∪ D) Solution
i. n(A)=7 ,n(B)=4, $n(A \times B)=28$
ii. n(C) =3. Number of relation from A to C = $2^{21}$
iii. False as one elements is mapping to two elements
iv. True
v. $B \times C= {(7,11),(7,13),(7,15) ,(9,11),(9,13),(9,15),(11,11),(11,13),(11,15),(13,11),(13,13),(13,15)}$
$C \times B= {(11,7),(11,9),(11,11) ,(11,13),(13,7),(13,9),(13,11) ,(13,13),(15,7),(15,9),(15,11) ,(15,13)}$
Mulitiple Choice questions
1. Given the relation R = {(6,4), (8,-1), (x,7), (-3,-6)}. Which of the following values for x will make relation R a function?
(a) 8
(b) 6
(c) -3
(d) 1 Solution
Solution (d)
2. Let n (A) = m, and n (B) = n. Then the total number of non-empty relations that can be defined from A to B is
(a) $m^n$
(b) mn- 1
(c) $2^{mn} -1$
(d) $n^m -1$ Solution
Solution (c)
3. The domain and range of real function f defined by f (x) = $\sqrt {x −1}$ is given by
(A) Domain = (1, ∞), Range = (0, ∞)
(B) Domain = [1, ∞), Range = (0, ∞)
(C) Domain = [1, ∞), Range = [0, ∞)
(D) Domain = [1, ∞), Range = [0, ∞) Solution
Solution (c)
4. If $f (x) = x^3 - \frac {1}{x^3}$ then $ f (x) + f(\frac {1}{x})$
(a) 0
1. What is the domain and range in the relation shown in below mapping Solution
Domain ={-2,2,4,5,6}
Range={4,16,25,36}
2. Please tell if the below mapping is function or not Solution
This is not function
Link type comprehension
There are two functions defined as below
Let P={(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)}
Q={(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)}
1.What is the domain and range of P
2.What is the domain and range of Q
3. What is the domain of function (Q-P)
4. List the ordered pair of (Q-P) in set notation
5. What is the domain of Q/P
6. List the ordered pair of (Q/P) in set notation Solution
1. Domain of P ={0,1,2,3,4,5}
Range of P ={5,4,3,2,1,0}
2. Domain of Q ={1,2,3,4,5,6}
Range of P ={1,4,9,16,25,36}
3. The domain of function (Q-P) is the intersection of domain of P and Q
So, Domain of (Q-P)={1,2,3,4,5}
4.(1,-3),( 2,1),(3,7),(4,15),(25)
5. The domain for Q/P is {1,2,3,4} as on 5 function p is zero
6. The ordered pair of Q/P
(1,1/4),(2,4/3),(3,9/2),(4,16)
