(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)=6x

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. $g(-1) = 6 (-1)^2 + 3 (-1) -2 =1$

(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=x

(12) The ordered pair {(x,y)| x=3 and y is real number} is a relation and function

- T
- F
- F
- F
- T
- T
- T
- T
- T
- F
- F
- F

(1) $y=x^2$

(2) $y=-|x|$

(3) $y=3x-7$

(4) $y=-x^4 + 3$

(5) $y= \sqrt {2-x}$

(6) $y =\frac {1}{ \sqrt {11-x}}$

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 $(0,\infty)$

(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)

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. $B \times C= {(7,11),(7,13),(7,15) ,(9,11),(9,13),(9,15),(11,11),(11,13),(11,15),(13,11),(13,13),(13,15)}$

$C \times B= {(11,7),(11,9),(11,11) ,(11,13),(13,7),(13,9),(13,11) ,(13,13),(15,7),(15,9),(15,11) ,(15,13)}$

(a) 8

(b) 6

(c) -3

(d) 1

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 (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 (c)

4. If $f (x) = x^3 - \frac {1}{x^3}$ then $ f (x) + f(\frac {1}{x})$

(a) 0

(b) $2x^3$

(c) 1

(d) $\frac {2}{x^3}$

$f (x) = x^3 - \frac {1}{x^3}$

$f(\frac {1}{x})) = \frac {1}{x^3} - x^3$

$ f (x) + f(\frac {1}{x}) = x^3 - \frac {1}{x^3} + \frac {1}{x^3} - x^3=0$

Domain ={-2,2,4,5,6}

Range={4,16,25,36}

2. Please tell if the below mapping is function or not

This is not function

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

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)

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