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Published on: 16/12/2019
Relations and Functions
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Questions + Answers key
Take MCQ Mathematics Test
1.
Find the domain of the function f defined by \(f(x)=\sqrt { 4-x } +\frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \)
2.
If f and g be two real function defined by \(f\left( x \right) =\sqrt { x+1 } \)and \(g\left( x \right) =\sqrt { 9-{ x }^{ 2 } } \). Then, describe each of the following functions. f+g
3.
Determine the range and domain of the relation: R = {(x, y): y = I x + 1|, x \(\in\) Z, IxI \(\le\)3}.
4.
Determine the domain and range of the relation: R = {(x, x2): x is prime, 10\(\le\) x \(\le\) 30}.
5.
Find x and y when (3x + 2y, x + 4y - 1) = (6, 6).
6.
A function f: R \(\rightarrow\)R is defined as
\(f(x)=\begin{cases} x^2+1\quad x\le-2 \\ 2x+1 \quad -2<x\le3\\ 2x^3-3\quad 3<x<8 \end{cases}\)
Find f(-3)
7.
A function f: R \(\rightarrow\)R is defined as
\(f(x)=\begin{cases} x^2+1\quad x\le-2 \\ 2x+1 \quad -2<x\le3\\ 2x^3-3\quad 3<x<8 \end{cases}\)
Find f(3)
8.
Find the domain and range of the following functions: f(x) =\({|x-3|\over x-3},x\in R\)
9.
Which of the following relations are functions? Give reasons if it is a function, and determine the domain and range {(3, 10), (5, 12), (7, 14), (5, 16)}
10.
Which of the following relations are functions? Give reasons if it is a function, and determine the domain and range{(4, 1), (6, 2), (8, 3), (10, 4), (12, 5)}
11.
Let A = {1, 2, 3}, B = {2, 3, 4, 5}, state as to which of the following sets R represents a relation of set A into set B?
If R represents a relation, write its domain and range.
R = {(1, 2), (2, 3), (3, 4), (3, 2)}
12.
Let A = {1, 2, 3}, B = {2, 3, 4, 5}, state as to which of the following sets R represents a relation of set A into set B?
If R represents a relation, write its domain and range.
R = {(2, 1), (1, 3), (4, 3)}
13.
If R is the relation "less than" from A ={1,2, 3, 4, 5} to B = {1, 4, 5}. Write down the set of ordered pairs corresponding to R. Find the inverse of R.
14.
Let A, B, C and D be any non-empty sets.
Prove that (A xB)\(\cap\)(CxD) =(A\(\cap\)C) x (B\(\cap\)D).
15.
The domain of the function \(f(x)=\sqrt{x-1}+\sqrt{3-x}\) is ______.
(1,\(\infty\))
(\(\infty\),5)
(1, 3)
[1, 3]
16.
If f(x) =\({x+1\over x-1}\) is a real function,\(\neq\) 1, then \(f[f\{f(2)\}]\) is ______.
-1
-3
3
4
17.
If \(f(x)={2^x+2^{-x}\over 2}\) then f(x+y) f(x-y)is equal to ______.
\({1\over2}[f(2x)+f(2y)]\)
\({1\over2}[f(2x)-f(2y)]\)
\({1\over3}[f(2x)+f(2y)]\)
\({1\over3}[f(2x)-f(2y)]\)
18.
If R is a relation from a finite set A having m elements to a finite set B having n elemen.ts, then the number of relations from A to B is ______.
2mn
2mn-1
2mn
mn
19.
Let R be a relation from a set A to B, then ______.
R=A\(\cup\)B
R =A\(\cap\)B
R\(\subset\)A x B
R \(\subset\) B x A
1.
Domain = \((-\infty ,-1)\cup (1,4)\)
2.
Domain \((f)\cap \) Domain \((g)=\left[ -1,3 \right] \)
\((f+g)(x)=\sqrt { x+1 } +\sqrt { 9-{ x }^{ 2 } } \)
3.
Domain = {-3, -2, -1,0, 1,2, 3};Range = {0,1,2,3, 4}
4.
Domain = {11, 13, 17, 19,23, 29}; Range = { 121, 169,289, 361, 529, 841}
5.
x = 1, y =\(3\over 2\)
6.
10
7.
7
8.
Domain = R - {3},Range = {1, - 1}
9.
No, because f(5) = 12 and f(5) = 16.
10.
Yes, Domain = {4, 6, 8,10, 12}, Range = {1, 2, 3,4, 5}
11.
Yes, {1, 2, 3}, {2, 3, 4}
12.
No
13.
HereA= {1, 2, 3, 4, 5} andB= {1, 4, 5},a\(\in\)A, b\(\in\)B.
\(\therefore\)a < b = 1 < 4, 1 < 5, 2 < 4, 2 < 5, 3 < 4, 3 < 5, 4 < 5
\(\therefore\)R = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3,5), (4, 5)}
Now R-1 = {(4, 1), (5, 1), (4, 2), (5, 2), (4, 3), (5, 3), (5, 4)}
14.
Let (x, y) \(\in\) (A x B) \(\cap\)(C x D).
\(\Rightarrow\)(x, y)\(\in\) A x B and (x, y) \(\in\) C x D
\(\Rightarrow\) [X\(\in\)A and y\(\in\)B] and [X\(\in\)CandY\(\in\)D]
\(\Rightarrow\) [x \(\in\)A and x\(\in\)C] and [y \(\in\) Band Y\(\in\) D]
\(\Rightarrow\)x \(\in\)(A \(\cap\) C) and Y\(\in\) (B\(\cap\)D)
\(\Rightarrow\) (x, y)\(\in\)(A\(\cap\)C) x (B\(\cap\)D)
\(\therefore\) (A x B)\(\cap\)(C x D)\(\subset\)(A \(\cap\)C) x (B\(\cap\)D) ...(i)
Let (x, y)\(\in\) (A\(\cap\)C) x (B\(\cap\)D)
\(\Rightarrow\) x\(\in\) A\(\cap\)C and Y \(\in\) B\(\cap\)D
\(\Rightarrow\) [x \(\in\) A and x \(\in\) C] and [y\(\in\) Band Y\(\in\) D]
\(\Rightarrow\) [x \(\in\) A and Y \(\in\) B] and [x\(\in\)C and Y\(\in\) D]
\(\Rightarrow\)(x, y) \(\in\) A x B and (x, y)\(\in\) C x D
\(\Rightarrow\)(x, y)\(\in\) (A x B)\(\cap\)(C x D)
\(\therefore\)(A\(\cap\)C) x (B\(\cap\)D)\(\subset\)(A x B)\(\cap\)(C x D) ...(ii)
From (i) and (ii), we have
(A x B)\(\cap\)(C x D) = (A\(\cap\)B) x (B\(\cap\)D)
15.
(d)
[1, 3]
16.
(c)
3
17.
(a)
\({1\over2}[f(2x)+f(2y)]\)
18.
(a)
2mn
19.
(c)
R\(\subset\)A x B
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