CBSE Questions for Class 12 Commerce Maths Relations And Functions Quiz 4 - MCQExams.com

If $$f:\left( 3,6 \right) \rightarrow \left( 1,3 \right) $$ is a function defined by $$\displaystyle f\left( x \right)=x-\left[ \frac { x }{ 3 }  \right] $$ $$($$ where $$\left[ . \right] $$ denotes the greatest integer function $$),$$ then $$\displaystyle f^{ -1 }\left( x \right)=$$
  • $$x-1$$
  • $$x+1$$
  • $$x$$
  • none of these
Let $$\displaystyle g(x)=1+x-[x]$$ and $$\displaystyle f(x)=\left\{\begin{matrix}{-1}\quad {x< 0} \\ {0} \quad {x=0}\\{1} \quad {x> 0} \end{matrix}\right.$$ Then for all  $$\displaystyle x, f\left \{ g\left ( x \right ) \right \}$$ is equal to 
  • $$x$$
  • $$1$$
  • $$\displaystyle f(x)$$
  • $$\displaystyle g(x)$$
The inverse of the function $$\displaystyle f(x) = \log_{2}(x+\sqrt{x^{2}+1}) $$ is
  • $$\displaystyle 2^{x}+2^{-x} $$
  • $$\displaystyle \frac{2^{x}+2^{-x}}{2}$$
  • $$\displaystyle \frac{2^{-x}-2^{x}}{2}$$
  • $$\displaystyle \frac{2^{x}-2^{-x}}{2}$$
If $$f : \{1,2,3,...\} \rightarrow \{0, \pm 1, \pm 2,...\}$$ is defined by
$$\displaystyle y=f(x)=\begin{cases} \displaystyle \frac { x }{ 2 } \quad \quad \text{ if x is even } \\ -\displaystyle \frac { (x-1) }{ 2 } \quad ,\text{ if x is odd } \end{cases}$$, then $$f^{-1}(100)$$ is
  • Function is not invertible as it is not onto
  • $$199$$
  • $$201$$
  • $$200$$
If $$\displaystyle f(y)=\frac{y}{\sqrt{1-y^2}}$$; $$\displaystyle g(y)=\frac{y}{\sqrt{1+y^2}}$$ then $$(fog)y$$ is equal to
  • $$\displaystyle \frac{y}{\sqrt{1-y^2}}$$
  • $$\displaystyle \frac{y}{\sqrt{1+y^2}}$$
  • $$y$$
  • $$2f(x)$$
If $$\displaystyle f(x)= (x-1)+(x+1)$$ and
$$\displaystyle g(x)= f\left \{ f(x) \right \}$$ then $$\displaystyle {g}'(3)$$
  • equals $$1$$
  • equals $$0$$
  • equals $$3$$
  • equals $$4$$
If $$\displaystyle f\left ( x \right )=x+\tan x$$ and $$\displaystyle g^{-1}=f$$ then $$\displaystyle g{}'\left ( x \right )$$ equals
  • $$\displaystyle \frac{1}{2+\left [ g\left ( x \right )+x \right ]^{2}}$$
  • $$\displaystyle \frac{1}{1+\left [ g\left ( x \right )-x \right ]^{2}}$$
  • $$\displaystyle \frac{1}{2+\left [ g\left ( x \right )-x \right ]^{2}}$$
  • $$\displaystyle \frac{1}{2-\left [ g\left ( x \right )-x \right ]^{2}}$$
Let $$f:[4,\infty )\rightarrow [4,\infty )$$ be a function defined by $$f\left( x \right)={ 5 }^{ x\left( x-4 \right)  }$$, then $$f^{ -1 }\left( x \right)$$ is
  • $$2-\sqrt { 4+\log _{ 5 }{ x } } $$
  • $$2+\sqrt { 4+\log _{ 5 }{ x } } $$
  • $$\displaystyle { \left( \frac { 1 }{ 5 } \right) }^{ x\left( x-4 \right) }$$
  • None of these
  • Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
  • Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  • Assertion is correct but Reason is incorrect
  • Assertion is incorrect but Reason is correct
Let $$\displaystyle f:N\rightarrow Y$$  be a function defined as $$f(x)=4x+3$$ where $$\displaystyle Y=\left \{ y \in N:y=4x+3 \right \}$$ for some $$\displaystyle x\in N$$ such that $$f$$ is invertible then its inverse is
  • $$\displaystyle g\left ( y \right )=4+\frac{y+3}{4}$$
  • $$\displaystyle g\left ( y \right )=\frac{2y-3}{4}$$
  • $$\displaystyle g\left ( y \right )=\frac{3y+4}{3}$$
  • $$\displaystyle g\left ( y \right )=\frac{y-3}{4}$$
If $$\displaystyle f\left ( x \right )=\left\{\begin{matrix}
x^{2}         x \geq 0\\
x              x < 0
\end{matrix}\right.$$
then $$\displaystyle (f o f)(x)$$ is given by
  • $$x^{2}$$ for $$x\geq 0$$ and $$x$$ for $$ x< 0$$
  • $$\displaystyle x^{4}$$ for $$\displaystyle x\geq 0$$ and $$x^{2}$$ for $$x< 0$$
  • $$ \displaystyle x^{4}$$ for $$ \displaystyle x\geq 0$$ and $$-x^{2} $$ for $$x < 0$$
  • $$\displaystyle x^{4}$$ for $$ x\geq 0$$ and $$x $$ for $$ x< 0$$
If $$\displaystyle f(x)= \frac{3x+2}{5x-3}$$ then
  • $$\displaystyle f^{-1}(x)= -f(x)$$
  • $$\displaystyle f^{-1}(x)= f(x)$$
  • $$\displaystyle fo(f(x))= -x $$
  • $$\displaystyle f^{-1}(x)= -\frac{1}{19}f(x)$$
Let $$\displaystyle f:R \rightarrow R $$ be defined as $$\displaystyle f(x)= x^{2}+5x+9$$ then $$\displaystyle f^{-1}(8) $$ equals to 
  • $$\displaystyle \left \{ \frac{-5+\sqrt{20}}{2},\frac{-5-\sqrt{21}}{2} \right \}$$
  • $$\displaystyle \left \{ \frac{-5+\sqrt{21}}{2} ,\frac{-5-\sqrt{21}}{2}\right \}$$
  • $$\displaystyle \left \{ \frac{5-\sqrt{21}}{2} ,\frac{21-\sqrt{5}}{2}\right \}$$
  • Does not exist.
Which of the following functions have inverse defined on the ranges
  • $$\displaystyle f\left ( x \right )=x^{2}, x $$ $$\displaystyle \in $$ R
  • $$\displaystyle f\left ( x \right )=x^{3},x$$ $$\displaystyle \in $$ R
  • $$\displaystyle f\left ( x \right )=e^{x},x$$ $$\displaystyle \in $$ R
  • $$\displaystyle f(x)=\sin x,$$ $$\displaystyle 0< x< 2\pi$$
Let f(x)=tan x, x$$\displaystyle \epsilon \left [ -\frac{\pi }{2},\frac{\pi }{2} \right ]$$ and $$\displaystyle g\left (x  \right )=\sqrt{1-x^{2}}$$ Determine $$g o f(1)$$.
  • 1
  • 0
  • -1
  • not defined
Find $$\displaystyle \phi \left [ \Psi \left ( x \right ) \right ]$$ and $$\displaystyle \Psi \left [ \phi \left ( x \right ) \right ]$$ if $$\displaystyle \phi \left ( x \right )=x^{2}+1$$ and $$\displaystyle \Psi \left ( x \right )=3^{x}.$$
  • $$\displaystyle \Psi \left [ \phi \left ( x \right ) \right ]=3^{x^{2}+1}.$$
  • $$\displaystyle \phi \left [ \Psi \left [ x \right ] \right ]=3^{2x}+1$$
  • $$\displaystyle \Psi \left [ \phi \left ( x \right ) \right ]=3^{x^{3}+1}.$$
  • $$\displaystyle \phi \left [ \Psi \left [ x \right ] \right ]=3^{x}+1$$
The inverse of the function$$\displaystyle f(x)=(1-(x-5)^{3})^{1/5} $$is 
  • $$5-(1-x^{5})^{1/3}$$
  • $$5+(1-x^{5})^{1/3}$$
  • $$5+(1+x^{5})^{1/3}$$
  • $$5-(1+x^{5})^{1/3}$$
If $$\displaystyle f\left ( x \right )=\frac{ax+b}{cx+d}$$ and $$\displaystyle \left ( fof \right )x=x,$$ then d=?
  • $$a$$
  • $$-a$$
  • $$b$$
  • $$-b$$
The inverse of the function $$\log_{e}x$$  is 
  • $$10^{x}$$.
  • $$e ^{x}$$
  • $$10^{e}$$.
  • $$x^{e}$$.
The total number of injective mappings from a set with $$m$$ elements to a set with $$n$$ elements,$$\displaystyle m\leq n,$$ is
  • $$\displaystyle m^{n}$$
  • $$\displaystyle n^{m}$$
  • $$\displaystyle \frac{n!}{\left ( n-m \right )!}$$
  • $$\displaystyle n!$$
If $$\displaystyle A= \left \{ a,b,c,d \right \}, B= \left \{ 1,2,3 \right \}$$ find whether or not the following sets of ordered pairs are relations from $$A$$ to $$B$$ or not.
$$\displaystyle R_{1}= \left \{ \left ( a,1 \right ), \left ( a,3 \right ) \right \}$$
$$\displaystyle R_{2}= \left \{ \left ( a,1 \right ), \left ( c,2 \right ), \left ( d,1 \right ) \right \}$$
$$\displaystyle R_{3}= \left \{ \left ( a,1 \right ), \left ( b,2 \right ), \left ( 3,c \right ) \right \}.$$
  • $$R_{1}$$ $$R_{2}$$ are relations but $$R_{3}$$ is not a relation.
  • $$R_{1}$$ $$R_{3}$$ are relations but $$R_{2}$$ is not a relation.
  • All are relations
  • none of these
Are the following sets of ordered pairs functions? If so, examine whether the mapping is surjective or injective :
{(x, y): x is a person, y is the mother of x}
  • injective (one- one ) and surjective (into)
  • injective (one- one ) and not surjective (into)
  • not injective (one- one ) and surjective (into)
  • not injective (one- one ) and not surjective (into)
Given $$\displaystyle f\left ( x \right )=\log \left ( \frac{1+x}{1-x} \right )$$ and $$\displaystyle g\left ( x \right )=\frac{3x+x^{3}}{1+3x^{2}}, fog (x)$$ equals
  • $$-f(x)$$
  • $$3f(x)$$
  • $$\displaystyle \left [ f\left ( x \right ) \right ]^{3}$$
  • none of these
Let $$R$$ be a relation from a set $$A$$ to a set $$B$$,then
  • $$\displaystyle R=A\cup B$$
  • $$\displaystyle R=A\cap B$$
  • $$\displaystyle R\subseteq A\times B$$
  • $$\displaystyle R\subseteq B\times A$$
If $$A=\{a,b,c,d\}, B=\{p,q,r,s\}$$, then which of the following are relations from $$A$$ to $$B$$? 
  • $$\displaystyle R_{1}= \left \{ \left ( a,p \right ), \left ( b,r \right ), \left ( c,s \right ) \right \}$$
  • $$\displaystyle R_{2}= \left \{ \left ( q,b \right ), \left ( c,s \right ), \left ( d,r \right ) \right \}$$
  • $$\displaystyle R_{3}= \left \{ \left ( a,p \right ), \left ( a,q \right ), \left ( d,p \right ), \left ( c,r \right ), \left ( b,r \right ) \right \}$$
  • $$\displaystyle R_{4}= \left \{ \left ( a,p \right ), \left ( q,a \right ), \left ( b,s \right ), \left ( s,b \right ) \right \}$$
If $$f:R\rightarrow R$$ and $$g:R\rightarrow R$$ are functions defined by $$f(x)=3x-1; g(x)=\sqrt{x+6}$$, then the value of $$(g\circ f^{-1})(2009)$$ is 
  • $$26$$
  • $$29$$
  • $$16$$
  • $$15$$
If $$\displaystyle X= \left \{ 1,2,3,4,5 \right \}, Y= \left \{ 1,3,5,7,9 \right \}$$ determine which of the following sets are mappings, relations or neither from A to B:
(i)$$\displaystyle F= \left \{ \left ( x,y \right ) \because y= x+2, x \in X, y \in Y \right \}$$
  • It is clearly a one-one onto mapping i.e. a bijection. It is also a relation.
  • It is clearly a many-one onto mapping. It is also a relation.
  • It is clearly a one-one but not onto mapping. It is also a relation.
  • It is not a mapping but a relation
Let $$f:[2,\infty)\rightarrow [1,\infty)$$defined by $$f(x)=2^{x^{4}-4x^{2}}$$ and $$\displaystyle g:\left[ \frac{\pi}{2},\pi \right] \rightarrow A$$ defined by $$ \displaystyle g(x)=\frac {\sin x+4}{\sin x-2}$$ be two invertible functions, then
$$f^{-1}(x)$$ is equal to
  • $$\sqrt{2+\sqrt{4-\log_{2}x}}$$
  • $$\sqrt{2+\sqrt{4+\log_{2}x}}$$
  • $$\sqrt{2-\sqrt{4+\log_{2}x}}$$
  • None of these
If $$f_{0}(x)\, =\, \dfrac{x}{(x\, +\, 1)}$$ and $$f_{n\, +\, 1}\, =\, f_{0}\circ f_{n}(x)$$ for $$n = 0, 1, 2,\cdots$$ then $$f_{n}(x)$$ is
  • $$\displaystyle \frac{x}{(n\, +\, 1) x\, +\, 1}$$
  • $$f_{0}(x)$$
  • $$\displaystyle \frac{nx}{nx\, +\, 1}$$
  • $$\displaystyle \frac{x}{nx\, +\, 1}$$
Let $$f(x)=x^{2}-2x$$ and $$g(x)=f(f(x)-1)+f(5-f(x)),$$ then
  • $$g(x)<0,\forall x\in R$$
  • $$g(x)<0$$ for some $$x\in R$$
  • $$g(x)\leq 0$$ for some $$x\in R$$
  • $$g(x)\geq 0,\forall x\in R$$
If $$f(x)=\begin{cases} 2x+3\quad \quad x\le 1 \\ a^{ 2 }x+1\quad x>1 \end{cases}$$, then the values of $$a$$ for which $$f(x)$$ is injective. 
  • $$-3$$
  • $$1$$
  • $$0$$
  • none of these
Which of the functions defined below are NOT one-one function(s) 
  • $$f(x)\, =\, 5(x^{2}\, +\, 4),\, (x\, \in\, R)$$
  • $$g(x)\, =\, 2x\, +\, \dfrac1x$$
  • $$h(x)\, =\, ln(x^{2}\, +\, x\, +\, 1)\,, (x\, \in\, R)$$
  • $$f(x)\, =\, e^{-x}$$
If $$g(x)=1+\sqrt { x } $$ and $$f(g(x))=3+2\sqrt { x } +x$$, then $$f(x)=$$
  • $$1+2{ x }^{ 2 }$$
  • $$2+{ x }^{ 2 }$$
  • $$1+x$$
  • $$2+x$$
Let $$X=\left\{ 1,2,3,4 \right\} $$ and $$Y=\left\{ 1,2,3,4 \right\} $$. Which of the following is a relation from $$X$$ to $$Y$$.
  • $${R}_{1}=\left\{ (x,y)| y=2+x, x\in X, y\in Y \right\} $$
  • $${R}_{2}=\left\{ (1,1),(2,1),(3,3),(4,3),(5,5) \right\} $$
  • $${R}_{3}=\left\{ (1,1),(1,3),(3,5),(3,7),(5,7) \right\} $$
  • $${R}_{4}=\left\{ (1,3),(2,5),(2,4),(7,9) \right\} $$
Find inverse $$f(x)=\log_{e}(x+\sqrt{x^{2}+1})$$
  • $$\sinh(x) $$
  • $$\cosh(x)$$
  • $$\tanh(x) $$
  • $$\coth(x)$$
Let f : {x,y,z} $$\rightarrow$$ {a,b,c} be a one-one function. It is known that only one of the following statment is true, and only one such function exists :

find the function f (as ordered pair).(i) f(x) $$\neq$$ b
(i) f(y) = b

(ii) f(z) $$\neq$$ a
  • {(x,b), (y,a), (z,c)}
  • {(x,a), (y,b), (z,c)}
  • {(x,b), (y,c), (z,a)}
  • {(x,c), (y,a), (z,b)}
Suppose f and g both are linear function with $$\displaystyle f(x)=-2x+1$$  and $$\displaystyle f \left ( g\left ( x \right ) \right )=6x-7$$ then slope of line $$y=g(x)$$ is
  • $$3$$
  • $$-3$$
  • $$6$$
  • $$-2$$
 from the given statement $$N$$ denotes the natural number and $$W$$ denotes the whole number, so which statement in the following is correct
  • N=W
  • N $$\subset$$ W
  • W $$\subset$$ N
  • N $$\cong$$ W
If $$f(x)=\begin{cases} x+1,\quad \quad if\quad x\, \leq \, 1 \\ 5-x^{ 2 }\quad \quad if\quad x>1 \end{cases},g(x)=\begin{cases} x\quad \quad if\quad x\leq 1 \\ 2-x\quad if\quad x>1 \end{cases}$$

and $$x\, \in\, (1, 2)$$, then $$g(f(x))$$ is equal to
  • $$x^{2}\, +\, 3$$
  • $$x^{2}\, -\, 3$$
  • $$5\, -\, x^{2}$$
  • $$1 - x$$
If $$g(x) = 2x + 1$$ and $$h(x) = 4x^{2} + 4x + 7$$, find a function $$f$$ such that $$f o g = h$$
  • $$f(x) = x^{3} - 6$$
  • $$f(x) = x^{2} + 6$$
  • $$f(x) = x^{2} - 6$$
  • $$f(x) = (2x+1)^2 + 6$$
Let $$X = \left\{1,2,3,4\right\}$$ and $$Y = \left\{1,3,5,7,9\right\}$$. Which of the following is relations from $$X$$ to $$Y$$
  • $$R_1 = \left\{(x,y) | y = 2x+1, x \in X, y \in Y\right\}$$
  • $$R_2 = \left\{(1,1),(2,1),(3,3),(4,3),(5,5)\right\}$$
  • $$R_3 = \left\{(1,1),(1,3),(3,5),(3,7),(5,7)\right\}$$
  • $$R_4 = \left\{(1,3),(2,5), (2,4), (7,9)\right\}$$
Which of the following are two distinct linear functions which map the interval $$[-1, 1]$$ onto $$[0, 2]$$
  • $$f(x) = 1 + x$$ or $$1 - x$$
  • $$f(x) = 1 + 2x$$ or $$1 - x$$
  • $$f(x) = 1 + x$$ or $$1 - 2x$$
  • $$f(x) = 1 + x$$ or $$2 - x$$
If $$f(x) =\ln {\displaystyle \frac { 1+x }{ 1-x }  } $$ and $$g(x)=\displaystyle \frac {3x+x^3}{1+3x^2}$$, then $$f[g(x)]$$ equals.

  • $$f(x)$$
  • $$[f(x)]^3$$
  • $$3f(x)$$
  • $${f(x)}^2$$
Let $$f(x) = e^{3x}, g(x) = \log_ex, x > 0$$, then $$fog (x)$$ is
  • $$3x$$
  • $$x^3$$
  • $$\log_{10}3x$$
  • $$\log3x$$
$$f(x)\, >\, x;\, \forall\, x\, \epsilon\, R.$$ The equation $$f (f(x)) -x = 0$$ has
  • Atleast one real root
  • More than one real root
  • No real root if f(x) is a polynomial & one real root if f(x) is not a polynomial
  • No real root at all
If $$ f : R \rightarrow R, f(x) = (x + 1)^2$$ and $$g : R \rightarrow  R, g(x) = x^2 + 1 $$ then $$(fog)(3)$$ is equal to
  • $$121$$
  • $$144$$
  • $$112$$
  • $$11$$
If $$f(x) = \sqrt{| x-1|}$$ and $$g(x) = \sin x$$, then $$(fog) (x)$$ equals
  • $$\sin \sqrt{| x-1|}$$
  • $$\left|\sin\dfrac{x}{2} - \cos\dfrac{x}{2}\right|$$
  • $$\left|\sin x + \cos x\right|$$
  • $$\left|\sin\dfrac{x}{2} + \cos\dfrac{x}{2}\right|$$
If $$f(x) = \log x$$, $$g(x) = x^3$$, then $$f[g(a)] + f[g(b)]$$ equals
  • $$f[g(a) + g(b)]$$
  • $$3f(ab)$$
  • $$g[f(ab)]$$
  • $$g[f(a) + f(b)]$$
If $$f(x) = x^3 $$ and $$g(x) = sin2x$$, then
  • $$g[f(1)] = 1$$
  • $$f(g(\pi/12) = 1/8$$
  • $$g{f(2)} = \sin 2$$
  • none of these
If $$f(x) = (a x^n)^{1/n},$$ where $$\ n \in N$$, then $$f\{f(x)\}$$ equals
  • $$0$$
  • $$x$$
  • $$x^n$$
  • none of these
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