CBSE Questions for Class 12 Commerce Maths Relations And Functions Quiz 7 - MCQExams.com

If $$f(x)=ax+b $$ and $$g(x)=cx+d$$, then $$f\left( g(x) \right) =g\left( f(x) \right) \Leftrightarrow$$
  • $$f(a)=g(c)$$
  • $$f(b)=g(b)$$
  • $$f(d)=g(b)$$
  • $$f(x)=g(a)$$
The inverse of the function $$f(x) = \log (x^{2} + 3x + 1), x\epsilon [1, 3]$$, assuming it to be an onto function, is
  • $$\dfrac {-3 + \sqrt {5 + 4e^{x}}}{2}$$
  • $$\dfrac {-3 \pm \sqrt {5 + 4e^{x}}}{2}$$
  • $$\dfrac {-3 - \sqrt {5 + 4e^{x}}}{2}$$
  • None of the above
Let $$f(x)={ x }^{ 3 }-3x+1$$. The number of different real solutions of $$f(f(x))=0$$
  • $$2$$
  • $$4$$
  • $$5$$
  • $$7$$
If $$f\left( x \right) $$ and $$g\left( x \right) $$ are two functions with $$g\left( x \right) =x-\dfrac { 1 }{ x } $$ and $$f\circ g\left( x \right) ={ x }^{ 3 }-\dfrac { 1 }{ { x }^{ 3 } } $$, then $$f^{ ' }\left( x \right) $$ is equal to
  • $$3{ x }^{ 2 }+3$$
  • $${ x }^{ 2 }-\dfrac { 1 }{ { x }^{ 2 } } $$
  • $$1+\dfrac { 1 }{ { x }^{ 2 } } $$
  • $$3{ x }^{ 2 }+\dfrac { 3 }{ { x }^{ 4 } } $$
 If $$f:R\rightarrow R$$, $$g:R\rightarrow R$$ are defined by$$ f(x)=5x-3$$, $$g(x)=x^{2}+3$$, then $$(gof^{-1})(3)$$=
  • $$\dfrac{25}{3}$$
  • $$ \dfrac{111}{25}$$
  • $$\dfrac{9}{25} $$
  • $$\dfrac{25}{111} $$
The inverse of the function $$y = 5^{\ln\ x}$$ is
  • $$x = y^{\frac {1}{\ln 5}}, y > 0$$
  • $$x = y^{\ln 5}, y > 0$$
  • $$x = y^{\frac {1}{\ln 5}}, y < 0$$
  • $$x = 5\ln y, y > 0$$
If $$f(x) = 2x^{3} + 7x - 5$$ then $$f^{-1}(4)$$ is
  • Equal to $$1$$
  • Equal to $$2$$
  • Equal to $$1/3$$
  • Non existent
$$f,g:R\rightarrow R$$ are functions such that $$f(x)=3x-\sin { \left( \cfrac { \pi x }{ 2 }  \right)  } ,g(x)={ x }^{ 3 }+2x-\sin { \left( \cfrac { \pi x }{ 2 }  \right)  } $$
The value of $$\cfrac { d }{ dx } { f }^{ -1 }{ \left( { g }^{ -1 }(x) \right)  }_{ x=12 }$$ is equal to
  • $$\cfrac { 2 }{ 30+x } $$
  • $$\cfrac { 2 }{ 30-x } $$
  • $$\cfrac { 2 }{ 3\left( 28-\pi \right) } $$
  • $$\cfrac { 2 }{ 3\left( 28+\pi \right) } $$
If $$f: R\rightarrow R$$ and $$g: R\rightarrow R$$ are defined $$f(x) = x - [x]$$ and $$g(x) = [x]\forall x\epsilon R, f(g(x))$$.
  • $$x$$
  • $$0$$
  • $$f(x)$$
  • $$g(x)$$
Let $$f : A \rightarrow B$$ be a function defined as $$f(x) = \dfrac {x - 1}{x - 2}$$, where $$A = R - \left \{2\right \}$$ and $$B = R - \left \{1\right \}$$. Then $$f$$ is :
  • invertible and $$f^{-1}(y) = \dfrac {2y + 1}{y - 1}$$
  • invertible and $$f^{-1}(y) = \dfrac {3y - 1}{y - 1}$$
  • not invertible
  • invertible and $$f^{-1}(y) = \dfrac {2y - 1}{y - 1}$$
If $$f(x)$$ is a real valued function, then which of the following is one-one function?
  • $$f(x)=e^{|x|}$$
  • $$f(x)=|e^x|$$
  • $$f(x)=\sin x$$
  • $$f(x)=|\sin x|$$
If $$A =\{1, 2, 3\}$$ and $$ B = \{4, 5\}$$ then the number of function $$f : A \rightarrow B$$ which is not onto is ______
  • $$2$$
  • $$6$$
  • $$8$$
  • $$4$$
If $$f\,: R \rightarrow R, g: R \rightarrow R\,$$ are defined by $$f(x)= 5x -3,g(x)=x^2 + 3$$, then, $$(gof^{-1})(3) =$$
  • $$\displaystyle \frac{25}{3}$$
  • $$\displaystyle \frac{111}{25}$$
  • $$\displaystyle \frac{9}{25}$$
  • $$\displaystyle \frac{25}{111}$$
Let $$f:A \to b$$ be a function defined by f(x) =$$\sqrt {1 - {x^2}} $$
  • f(x) is one-one if A =[0,1]
  • f(x) is onto if B = [0,1]
  • f(x) is one-one if A =[-1 , 0]
  • f(x) is onto if B = [-1,1]
If $$f:R\rightarrow R,f(x)=\begin{cases} 1\quad \quad x>0 \\ 0\quad \quad x=0 \\-1\quad x<0 \end{cases}$$ and $$g:R\rightarrow R,g(x)=\left[ x \right] $$, then $$\left( f\circ g \right) \left( \pi  \right)$$ is:
  • $$\pi$$
  • $$0$$
  • $$1$$
  • $$-1$$
The inverse of the function $$y=\large{\frac{e^x-e^{-x}}{e^x+e^{-x}}}$$ is
  • $$\large{\frac{1}{2}}$$ $$\log \large{\frac{1+x}{1-x}}$$
  • $$\large{\frac{1}{2}}$$ $$\log \large{\frac{2+x}{2-x}}$$
  • $$\large{\frac{1}{2}}$$ $$\log \large{\frac{1-x}{1+x}}$$
  • $$2 \log (1+x)$$
Let $$f\left( x \right) = {x^2}$$ and $$g\left( x \right) = \sqrt x $$ (where $$x > 0$$),then
  • $$f\left( {g\left( x \right)} \right) = x$$
  • $$g\left( {f\left( x \right)} \right) = x$$
  • The least value of $$f\left( {g\left( x \right)} \right) + {1 \over {g\left( {f\left( x \right)} \right)}}$$ is $$2$$
  • The least value of $$g\left( {f\left( x \right)} \right) + {1 \over {f\left( {g\left( x \right)} \right)}}$$ is $$2$$
The solution of  $$( 3 * 4) * 3$$,  when $$*$$ is a binary operation on Z such that: $$a * b = a + b$$, is.
  • $$10$$
  • $$-10$$
  • $$-\dfrac16$$
  • $$-6$$
If $$g\left( x \right) = {x^2} + x - 2$$ and $$\frac{1}{2}gof\left( x \right) = 2{x^2} + 5x + 2$$, then $$f\left( x \right)$$ is
  • $$2x-3$$
  • $$2x+3$$
  • $$2{x^2} + 3x + 1$$
  • $$2{x^2} - 3x -1$$
Let f : $$R \to R$$ and g : $$R \to R$$ be two one-one and onto functions such that they are the mirror images of each other about the line y =If h(x) = f(x) + g(x), then h(0) equal to
  • 2
  • 4
  • 0
  • 1
If $$f: A\rightarrow A$$ defined by $$f(x) =\dfrac{4x +3}{6x-4}$$ where $$A= R-\dfrac{2}{3}$$. Find $$f^{-1}$$
  • $$2x$$
  • $$\dfrac {4x+3}{6x-4}$$
  • $$\dfrac x2$$
  • None of these
If the binary operation $$*$$ is defined on a set of integers as $$a * b  = a + 3  b ^{2} $$ , then the value of $$2 * 3$$ is
  • $$27$$
  • $$29$$
  • $$2$$
  • None of these
Let $$f$$, $$g:R\rightarrow {R}$$ be two functions defined as $$ f\left( x \right) =\left| x \right| +x$$, $$ g\left( x \right) =\left| x \right| -x, \forall x\in R$$. Then, find $$fog(x)$$ 
  • $$||x|-x|-|x|-x$$
  • $$||x|-x|+|x|-x$$
  • $$||x|-x|-|x|+x$$
  • None of thesse
Consider set $$A={1,2,3,4}$$ and set $$B={0,2,4,6,8}$$, then the number of one-one function from set $$A$$ to set $$B$$ is ?
  • $$5$$
  • $$24$$
  • $$120$$
  • None of these
The function $$*$$ on $$N$$ as
 $$a * b = ( a - b)^{2} $$ is a binary operator 
  • True
  • False
 If the binary operation $$*$$ is on set of  integers $$Z$$ is defined as
$$a * b = a + 2b ^{2}$$ , then the value of $$(8 * 3) * 2$$
  • $$26$$
  • $$22$$
  • $$32$$
  • $$34$$
If $$f(x)=2x+5$$ and $$g(x)=x^2+1$$ be two real function , then value of $$fog$$ at x=1
  • $$9$$
  • $$6$$
  • $$5$$
  • $$4$$
If $$g(f(x) ) = |\sin x |$$ and $$f(g(x))=(\sin\sqrt x)^2$$ , then 
  • $$f(x) = \sin^2x. g(x) =\sqrt x$$
  • $$f(x) = \sin x , g(x) =|x| $$
  • $$f(x) = x^2, g(x) = \sin \sqrt x$$
  • f and g can not be determined
Let $$f : R \rightarrow R$$ be defined by $$f(x) = x^2 - 3x + 4 $$ for all $$x \epsilon R$$, then $$f^{-1}(2)$$ is 
  • 2
  • 1
  • Not defined
  • $$\dfrac{1}{2}$$
Let $$f(x+\dfrac{1}{x})=x^2+\dfrac{1}{x^2}(x\neq 0)$$, then $$f(x)=$$
  • $$x^2$$
  • $$x^2-1$$
  • $$x^2-2$$
  • N.O.T
The set onto which the derivative of the function $$f(x)=x(\log x-1)$$ maps the range $$[1,\infty )$$ is
  • $$\left[1,\infty \right)$$
  • $$\left( e,\infty \right)$$
  • $$\left[e,\infty \right)$$
  • $$\left( 0,0 \right)$$
Let $$E=\{1, 2, 3, 4\}$$ and $$F=\{1, 2\}$$ then the number of onto functions from E to F is
  • $$14$$
  • $$16$$
  • $$12$$
  • $$8$$
Let $$f\left( x \right) ={ x }^{ 2 },g\left( x \right) ={ 2 }^{ x }$$, then solution set of $$fog\left( x \right) =gof\left( x \right) $$ is
  • R
  • $$\left\{ 0 \right\} $$
  • $$\left\{ 0,2 \right\} $$
  • None of these
If $$f\left( x \right) =\begin{cases} 2+x,\quad x\ge 0 \\ 2-x,\quad x<0 \end{cases}$$ then $$f\left( f\left( x \right)  \right) $$ is given by
  • $$f\left( f\left( x \right) \right) =\begin{cases} 2+x,\quad x\ge 0 \\ 4-x,\quad x<0 \end{cases}$$
  • $$f\left( f\left( x \right) \right) =\begin{cases} 2+x,\quad x\ge 0 \\ 2-x,\quad x<0 \end{cases}$$
  • $$f\left( f\left( x \right) \right) =\begin{cases} 4+x,\quad x< 0 \\ x,\quad x\ge 0 \end{cases}$$
  • $$f\left( f\left( x \right) \right) =\begin{cases} 4+x,\quad x\ge 0 \\ x,\quad x<0 \end{cases}$$
Let $$f\left[-1, \dfrac{-1}{2}\right] \to [-1, 1]$$ is defined by $$f(x) = 4x^3 - 3x$$, then $$f^{-1} (x) =$$ ____ .
  • $$\cos \left(\dfrac{1}{3}\cos^{-1} x\right)$$
  • $$\cos \left(3\cos^{-1} x\right)$$
  • $$\sin \left(\dfrac{1}{3}\sin^{-1} x\right)$$
  • $$\cos \left(\dfrac{2\pi}{3}+\dfrac{1}{3} \cos^{-1} x\right)$$
If : $$f(x) = 5 {x}^{2}$$, $$g(x) = 3x^{4}$$, then : $$(fog) (-1) =$$ 
  • $$45$$
  • $$-54$$
  • $$-32$$
  • $$-64$$
Let $$f:X \to \left[ {1,\,27} \right]$$ be  a function by $$f\left( x \right) = 5\sin x + 12\cos x + 14$$. The set $$X$$ so that $$f$$ is one-one and onto is 
  • $$\left[ { - \pi /2,\pi /2\,} \right]$$
  • $$\left[ {0,\,\pi } \right]$$
  • $$\left[ {0,\,\pi /2} \right]$$
  • non of these
For $$a,\ b\ \in \ R-\left\{ 0 \right\}$$, let $$f(x)=ax^{2}+bx+a$$ satisfies $$f\left(x+\dfrac{7}{4}\right)=f\left(\dfrac{7}{4}-x\right) \forall \ x\ \in\ R$$.
Also the equation $$f(x)=7x+a$$ has only one real distinct solution. The minimum value of $$f(x)$$ in $$\left[0,\dfrac{3}{2}\right]$$ is equal to
  • $$\dfrac{-33}{8}$$
  • $$0$$
  • $$4$$
  • $$-2$$
If $$f\left( x \right) = (1 - x)$$ , $$x \in \left[ { - 3,3} \right]$$ , then the domain of $$f\left( {f\left( x \right)} \right)$$ is
  • $$\left[ { - 2,3} \right]$$
  • $$\left( { - 2,3} \right)$$
  • $$\left[ { - 2,3]} \right.$$
  • $$( - 2,3]$$
If f(g(x))=5x+2 and g(x)=8x then f(x)=
  • $$ \frac{5}{8}x+2$$.
  • $$ \frac{8}{5}x+2$$.
  • $$ \frac{5}{8}x-2$$.
  • 8x-2
  • 5x-2
Let $$g\left( x \right) =1+x-\left[ x \right] $$ and $$f\left( x \right) =\begin{cases} -1,x<0 \\ 0,x=0 \\ 1,x>0 \end{cases}$$ Then for all $$x,f\left( g\left( x \right) \right) $$ is equal to (where $$\left[ . \right] $$ represents the greatest integer function)
  • $$x$$
  • $$1$$
  • $$f\left( x \right)$$
  • $$g\left( x \right)$$
Let $$f:(2,3) \rightarrow (0,1)$$ be defined by $$f(x)=x-[x]$$ then $$f^{-1}(x)$$ equals

  • $$x-2$$
  • $$x+1$$
  • $$x-1$$
  • $$x+2$$
If $$f(x)=\frac{x}{\sqrt{1-x^{2}}}$$ and g(x) = $$f(x)=\frac{x}{\sqrt{1+x^{2}}}$$ , then (fog)(x) =
  • $$f(x)=\frac{x}{\sqrt{1-x^{2}}}$$
  • $$f(x)=\frac{x}{\sqrt{1+x^{2}}}$$
  • $$x^{2}$$
  • x
Let $$f:X\rightarrow Y$$ be an invertible  function. Then f has unique inverse.
  • True
  • False
If $$f:R \rightarrow R, f(x)=2x-1$$ and $$g; R \rightarrow R, g(x)=x^{2}+2$$, then $$(gof)(x)$$ equals-
  • $$2x^{2}-1$$
  • $$(2x-1)^{2}$$
  • $$2x^{2}+3$$
  • $$4x^{2}-4x+3$$
Let $$f(x)=\dfrac{1}{x^{2}}$$ for $$x\ge 1$$, and $$g(x)$$ is its reflection in the line mirror $$y=x$$, then function $$h(x)=\begin{cases} f\left( x \right)  & x\ge 1 \\ g\left( x \right)  & 0<x<1 \end{cases}$$, is
  • derivable at $$x=1$$
  • continuous at $$x=1$$
  • not derivable at $$x=1$$
  • not continuous at $$x=1$$
If $$f(x)=\begin{cases} x+1 & x\epsilon \left[ -1,0 \right]  \\ { x }^{ 2 }+1 & x\epsilon \left( 0,1 \right)  \end{cases}$$, then the value of $$\dfrac{f^{-1}(0)+f^{-1}(1)+f^{-1}(2)}{f(-1)+f(0)+f(1)}$$ is-
  • $$0$$
  • $$1$$
  • $$2$$
  • $$\dfrac{1}{3}$$
The last three digits, if $$(12345956)_{10}$$ is expressed in binary system.
  • $$110$$
  • $$210$$
  • $$100$$
  • $$010$$
If $$ f ( x ) = \left( a - x ^ { n } \right) ^ { 1 / n }$$ where $$ a > 0$$ and } $$n$$ is a positive integer then$$( f o f ) ( x )$$ is
  • $$f ( x )$$
  • x
  • 0
  • 1
if $$f\left( x \right) = \log \left( {\dfrac{{1 +x}}{{1 - x}}} \right)$$ and $$g\left( x \right) = \dfrac{{3x + {x^3}}}{{1 + 3{x^2}}}$$ then $$\left( {f(g(x)))} \right)$$ is equal to
  • $$ - f\left( x \right)$$
  • $$3f\left( x \right)$$
  • $${\left( {f\left( x \right)} \right)^3}$$
  • $$f\left( {3x} \right)$$
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