CBSE Questions for Class 12 Commerce Maths Relations And Functions Quiz 10 - MCQExams.com

Let $$f(x) = \begin{cases} 1+x, & 0\leq x\leq 2 \\ 3-x, & 2<x\leq 3 \end{cases}$$, then find $$(fof)(x)$$
  • $$\left\{\begin{matrix}2 + x, &0 \leq x \leq 1 \\ 2 - x, &1 < x \leq 2 \\ 4 - x, &2 < x \leq 3 \end{matrix}\right.$$
  • $$\left\{\begin{matrix}2 - x, &0 \leq x \leq 1 \\ 2 + x, &1 < x \leq 2 \\ 4 - x, &2 < x \leq 3 \end{matrix}\right.$$
  • $$\left\{\begin{matrix}2 + x, &0 \leq x \leq 1 \\ 2 - x, &1 < x \leq 2 \\ 4 + x, &2 < x \leq 3 \end{matrix}\right.$$
  • None of these
$$f(x)\, =\, -1\, + |x\, -\, 2|, \, 0\, \leq\, x\, \leq\, 4$$
$$g(x)\, =\, 2\, -\, |x|,\, -1\, \leq\, x\, \leq\, 3$$
Which of the following is true
  • $$fog(x)\, =\, \begin{cases} -(1\, +\, x), & & -1\, \leq\, x\, \leq\, 0\\ x + 1 & & 0\, <\, x\, \leq\, 2 \end{cases}$$
  • $$gof(x)\, =\, \begin{cases}x\,+\,1, & 0\, \leq\, x\, <\, 1\\ 3\, -\, x, & 1\, \leq\, x\, \leq\, 2 \\ x\, -\, 1, & 2\, <\, x\, \leq\, 3\\ 5\, -\, x, & 3\, <\, x\, \leq\, 4\end{cases}$$
  • $$fog(x)\, =\, \begin{cases} -(1\, +\, 2x), & & -1\, \leq\, x\, \leq\, 0\\ x - 1 & & 0\, <\, x\, \leq\, 2 \end{cases}$$
  • $$gof(x)\, =\, \begin{cases}x\,+\,1, & 0\, \leq\, x\, <\, 1\\ 3\, -\, x, & 1\, \leq\, x\, \leq\, 2 \\ x\, +\, 1, & 2\, <\, x\, \leq\, 3\\ 5\, -\, x, & 3\, <\, x\, \leq\, 4\end{cases}$$
Find the inverse of the quadratic function $$f$$.
$$f(x)=-{x}^{2}+2, x>=0$$
  • $$\sqrt{(2+x)}$$
  • $$(2-x)$$
  • $$\sqrt{(2+x^2)}$$
  • $$\sqrt{(2-x)}$$
If f(x)=x+5 and g(x)=$$\displaystyle \sqrt{x^{2}-9}$$  then find the domain of gof(x)
  • (-8,-2)
  • $$\displaystyle \left ( -\infty ,-8 \right )\cup \left ( -2,\infty \right )$$
  • $$\displaystyle (-\infty ,-8]\cup [-2,\infty )$$
  • $$\displaystyle (-(-\infty ,-8]\cup [-2,\infty )$$
Let $$f(x)=\ln x$$  and  $$g(x)\, =\, \left (\displaystyle \frac{x^{4}\, -\, x^{3}\, +\, 3x^{2}\, -\, 2x\, +\, 2}{2x^{2}\, -\, 2x\, +\, 3 )}\right )$$. The domain of $$f(g(x))$$ is
  • $$(-\, \infty,\, \infty)$$
  • $$[0,\, \infty)$$
  • $$(0,\, \infty)$$
  • $$[1,\, \infty)$$
If $$f(x)=\begin{cases} x+1,\quad \quad if\quad x\, \leq \, 1 \\ 5-x^{ 2 }\quad \quad if\quad x>1 \end{cases},g(x)=\begin{cases} x\quad \quad if\quad x\leq 1 \\ 2-x\quad if\quad x>1 \end{cases}$$
Number of negative integral solutions of $$g(f(x)) + 2 = 0$$ are 
  • $$0$$
  • $$3$$
  • $$1$$
  • $$2$$
Find the inverse of the exponential function $$f$$.
$$f(x)={e}^{x-1}+3$$
  • $$\ln(x-1)+3$$
  • $$\ln(x-3)+1$$
  • $$\ln(x-1)$$
  • $$\ln(x-2)-3$$
Given two functions $$f(x)$$ and $$g(x)$$ such that $$f(x) = \sin (arctan x), g(x) =\tan (arc\sin x)$$, and $$0\leq x < \dfrac {\pi}{2}$$. The value of the composite function $$f\left (g\left (\dfrac {\pi}{10}\right )\right ) $$ is:
  • $$0.314$$
  • $$0.354$$
  • $$0.577$$
  • $$0.707$$
  • $$0.866$$
If $$f(x) = x^{2} + x$$ and $$g(x) = \sqrt {x}$$, then the value of $$f(g(3))$$ is
  • $$1.73$$
  • $$3.46$$
  • $$4.73$$
  • $$7.34$$
  • $$12.00$$
Find the inverse of the logarithmic function $$f$$.
$$f(x)=\ln(x+2)-3$$
  • $${e}^{x+3}-2$$
  • $${e}^{x+1}-2$$
  • $${e}^{x+2}-3$$
  • $${e}^{x-2}+3$$
If $$f(x)=2x^3$$ and $$g(x)=3x$$, calculate the value of $$g(f(-2))-f(g(2))$$.
  • $$-480$$
  • $$-384$$
  • $$0$$
  • $$384$$
  • $$480$$
Find $$g(x)$$, if $$f(x) = 5x^{2} + 4$$ and $$f(g(3)) = 84$$
  • $$3x - 10$$
  • $$4x - 7$$
  • $$6x - 17$$
  • $$x^{2} - 5$$
  • $$x^{2} - 3$$
Find the inverse of the cube root function $$f$$.
$$f(x)={(x+1)}^{1/3}$$
  • $${x}^{3}+1$$
  • $${x}^{3}-1$$
  • $${x}-1$$
  • $${x}+1$$
Find the inverse of the logarithmic function $$f$$.
$$f(x)=\ln(x)$$
  • $${e}^{x}$$
  • $$-{e}^{x}$$
  • $${e}^{2x}$$
  • $$-{e}^{2x}$$
Find the inverse of the square root function $$f$$.
$$f(x)=\sqrt{x-1}$$
  • $$x^2-1$$
  • $$x^2+1$$
  • $$x+1$$
  • $$x-1$$
If the binary operation $$a$$ # $$b = a^{b} - \sqrt {b}$$, then $$(2$$ # $$4) - (4$$ # $$2) =$$
  • $$-32$$
  • $$\sqrt {2} - 2$$
  • $$0$$
  • $$\sqrt {2} - 2$$
  • $$32$$
Find the maximum value of $$g(f(x))$$ if:
$$f(x) = x + 4$$ and
$$g(x) = 6 - x^{2}$$
  • $$-6$$
  • $$-4$$
  • $$2$$
  • $$4$$
  • $$6$$
If $$h(x) = x^2, g(x)= x^2 -3$$ and f(x)= x -2, what can you say about ho(gof) and (hog)of?
  • (hog)of $$\neq $$ ho(gof)
  • ho(gof) = (hog)of
  • (hog)of = 4 ho(gof)
  • ho(gof)=$$(x^2-4x-1)^2$$
Which of the following functions are not identical?
  • $$f(x)=\frac{x}{x^2}$$ and $$g(x) = \frac{1}{x}$$
  • $$f(x)=\frac{x^2}{x}$$ and $$g(x) =$$
  • $$f(x)=In \,x^4$$ and g(x)= 4 In Xx
  • f(x) = In {(x-1)(x-2)} and g(x) = In (x-2)+In (x-3)
If f(x)=x+2and $$g(x)=x^2-3$$, then which is true?
  • fog $$\neq $$ gog
  • 2fog = gof
  • fog = gof
  • fog = 2 gof
Let $$f:\{x, y , z\} \rightarrow \{1, 2, 3\}$$ be a one-one mapping such that only one of the following three statements and remaining two are false : $$f(x) \neq 2, f(y) =2, f(z) \neq 1$$, then 
  • $$f(x) > f(y) > f(z)$$
  • $$f(x) < f(y) < f(z)$$
  • $$f(y) < f(y) < f(z)$$
  • $$f(y) < f(z) < f(x)$$
$$f(x)=x^2-x+5$$ and $$g(x) = f^{-1}(x)$$. Then $$g'(7)=$$
  • $$1$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{13}$$
  • $$\dfrac{1}{4}$$
Let $$f: R\rightarrow R$$ be a function such that $$f(x) = ax + 3\sin x + 4\cos x$$. Then $$f(x)$$ is invertible if
  • $$a\in (-5,5)$$
  • $$a\in (-\infty,5)$$
  • $$a\in (-5,\infty)$$
  • None of the above
In the set $$Q^{+}$$ of all positive rational numbers, the operation $$\ast$$ is defined by the formula $$a\ast b = \dfrac {ab}{6}$$. Then, the inverse of $$9$$ with respect to $$\ast$$ is
  • $$4$$
  • $$3$$
  • $$\dfrac {1}{9}$$
  • $$\dfrac {1}{3}$$
The function f is defined as $$f=\{(x,y)|y=\frac{2x+1}{x-3}$$ where $$x\neq 3\}$$. Find the value of K so that the inverse of f will be $$f^{-1}=\{(x, y)|y=\frac{3x+1}{x-k}$$ where $$x\neq K\}$$
  • 1
  • 2
  • 3
  • 4
  • 5
If $$f(x) = 4x^{2} - 1$$ and $$g(x) = 8x + 7, g\circ f(2) =$$
  • $$15$$
  • $$23$$
  • $$127$$
  • $$345$$
  • $$2115$$
Consider set $$A={1,2,3,4}$$ and set $$B={0,2,4,6,8}$$, then the number of one-one function set $$A$$ to set $$B$$ in which $$f(i)\neq i$$ is,
  • $$84$$
  • $$78$$
  • $$42$$
  • $$24$$
Let $$f : N \rightarrow N$$, Where $$f(x) = x+(-1)^{x-1}$$, then the inverse of $$f$$ is
  • $$f^{-1}(x) = x + (-1)^{x-1}, x \in N$$
  • $$f^{-1}(x) = 3x + (-1)^{x-1}, x \in N$$
  • $$f^{-1}(x) = x , x \in N$$
  • $$f^{-1}(x) = (-1)^{x-1}, x \in N$$
If $$f:R\rightarrow S$$ defined by
$$f(x)=4\sin { x } -3\cos { x } +1$$ is onto, then $$S$$ is equal to
  • $$[-5,5]$$
  • $$(-5,5)$$
  • $$(-4,6)$$
  • $$[-4,6]$$
$$f:\left( 0,\infty  \right) \rightarrow \left( 0,\infty  \right) $$ is defined by $$f(x)=\begin{cases} { 2 }^{ x },\quad x\in \left( 0,1 \right)  \\ { 5 }^{ x },\quad x\in [1,\infty ) \end{cases}$$ is
  • one-one but not onto
  • onto but not one-one
  • neither one-one nor onto
  • bijective
Let for $$a \neq a_{1} \neq 0,\ f(x)=ax^{2}+bx+c,\ g(x)=a_{1}x^{2}+b_{1}x+c_{1}$$ and $$p(x)=f(x)-g(x)$$. If $$p(x)=0$$ only for $$x=-1$$ and $$p(-2)=2$$, then the value of $$p(2)$$ is
  • $$6$$
  • $$18$$
  • $$3$$
  • $$9$$
Given that $$f'(x) > g'(x)$$ for all real x, and $$f(0) = g(0)$$. Then $$f(x) < g(x)$$ for all x belongs to
  • $$(0, \infty)$$
  • $$(- \infty, 0)$$
  • $$(- \infty, \infty)$$
  • none of these
Let $$f(x) = \dfrac {x}{1 - x}$$ and let $$\alpha$$ be a real number. If $$x_{0} = \alpha, x_{1} = f(x_{0}), x_{2} = f(x_{1}), ....$$ and $$x_{2011} = - \dfrac {1}{2012}$$ then the value of $$\alpha$$ is
  • $$\dfrac {2011}{2012}$$
  • $$1$$
  • $$2011$$
  • $$-1$$
If $$ \phi (x) = 3
f(\frac{x^2}{3} ) + f(3-x^2) \forall x \in (3,4)$$ where  $$f(x) >0 \forall  x (-3,4)$$ then $$\phi (x)$$ is ____________.
  • (a) increasing in $$( \frac{3}{2} ,4)$$
  • (b) decreasing in $$( 3, \frac{3}{2} )$$
  • (c) increasing $$( -\frac{3}{2} , 0)$$
  • decreasing in $$( 0, \frac{3}{2})$$
Let $$f: R\rightarrow R; f(x)=2x^3-3(p+2)x^2+12px-7$$, $$-5\leq p \leq 5$$, $$p\in I$$. Then the number of values of p for $$f(x)$$ to be invertible is?
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
If $$f(0)=5$$, then minimum possible number of values of x satisfying $$f(x)=5$$, for $$x\in [0, 170]$$ is?
  • $$21$$
  • $$12$$
  • $$11$$
  • $$22$$
Let f(x) and g(x) be the differentiable functions for $$1\le x\le 3$$ such that f(1)=2=g(1) and f(3)=Let there exist exactly one real number $$cE (1,3)$$ such that 3f'(c)=g'(c), then the value of g(3) must be
  • 12
  • 13
  • 16
  • 26
Let $$f\left( x \right)={2^{10}}x + 1$$ and $$g\left( x \right) = {3^{10}}x - 1$$ , If $$\left( {fog} \right)\left( x \right) = x$$ , then $$x$$ is equal to
  • $$\dfrac{{{3^{10}} - 1}}{{{3^{10}} - {2^{ - 10}}}}$$
  • $$\dfrac{{{2^{10}} - 1}}{{{2^{10}} - {3^{ - 10}}}}$$
  • $$\dfrac{{1 - {3^{10}}}}{{{2^{10}} - {3^{ - 10}}}}$$
  • $$\dfrac{{1 - {2^{-10}}}}{{{3^{10}} - {2^{ - 10}}}}$$
A real valued function $$f(x)$$ satisfies the function equation $$f(x-y)=f(x)f(y)-f(a-x)f(a+y)$$ where a is a given constant and $$f(0)=1, f(2a-x)$$ is equal to?
  • $$f(a)+f(a-x)$$
  • $$f(-x)$$
  • $$-f(x)$$
  • $$f(x)$$
If $$f(x)=|x|$$ and $$g(x)=[x]$$, then value of $$fog \left(-\dfrac {1}{4}\right)+gof \left(-\dfrac {1}{4}\right)$$ is  ?
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$1/4$$
The function $$f : R\rightarrow R$$ given by, then $$f(x)=3-2\sin x$$ is
  • one-one
  • onto
  • bijective
  • None of these

The inverse of the function $$f\left( x \right) = \frac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}} + 2$$ is given by

  • $$\log {\left( {\frac{{x - 2}}{{x - 1}}} \right)^{1\backslash 2}}$$
  • $$\dfrac{1}{2}\log_e {\left( {\frac{{x - 1}}{{3-x}}} \right)}$$
  • $$\eqalign{& \log {\left( {\frac{x}{{2 - x}}} \right)^{1\backslash 2}} \cr & \cr} $$
  • $$\log {\left( {\frac{{x - 1}}{{3 - x}}} \right)^{1\backslash 2}}$$
If the function $$ f : R \rightarrow R $$ be such that $$ f ( x ) = x - [ x ] $$ where $$[ . ]$$ denotes the greatest integer less than or equal to $$ x $$ then $$ f ^ { - 1 } ( x ) $$ is 
  • $$ \frac { 1 } { x - [ x ] } $$
  • $$ \frac { 1 } { x } - \frac { 1 } { x } $$
  • Not defined
  • $$ x - [ x ] $$
If $$f(x)=x-\cfrac{1}{x}$$ then number of solutions of $$f(f(f(x)))=1$$ is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
Show that the function $$f:[0, \infty)\rightarrow [0, \infty)$$ defined by $$f(x)=\dfrac{2x}{1+2x}$$ is?
  • One-one and onto
  • One-one but not onto
  • Not one-one but onto
  • Neither one-one nor onto
If $$f(x)=1+|x-1|,-1 \le x \le 3$$ and $$g(x)=2-|x+1|,-2 \le x \le 2$$ then choose the appropriate option.
  • $$fog(x)=x-1$$ for $$x\ \in\ (0,1)$$
  • $$fog(x)=x$$ for $$x\ \in\ (-1,1)$$
  • $$gog(x)=x$$ for $$x\ \in\ (-1,2)$$
  • $$all\ of\ these$$
Let $$f(x)=\dfrac{x^{2}-4}{x^{2}+4}$$ for $$|x|>2$$, then the function $$f:(-\infty, -2)\cup [2,\infty)\rightarrow (-1,1)$$ is
  • One-one into
  • One-one onto
  • Many one into
  • Many one onto
If $$f(x)=ax+b$$ and $$f(f(f(x)))=27x+13$$ where a and b are real numbers, then-
  • a+b=3
  • a+b=4
  • f'(x)=3
  • f'(x)=-3
If $$f\left( x \right)  = \sin ^{ 2 }{ x } + \sin ^{ 2 }({ { x }+\frac { \pi  }{ 3 })  + \cos { x\cos { \left( { x } + \frac { \pi  }{ 3 }  \right),  ~g(\frac { 5 }{ 4 })  = 1, \text{then} \left( gof \right)\left( x \right)  }  }\ \text{ is}\  \text{equal}\  \text{to}  } $$
  • $$1$$
  • $$0$$
  • $$\frac{1}{4}$$
  • $$\frac{1}{2}$$
$$f:A \rightarrow A,A=\left\{a_{1},a_{2},a_{3},a_{4},a_{5}\right\}$$, then the number of one one function so that $$f(x_{i})\neq x_{i},x_{i}\ \in\ A$$ is
  • $$44$$
  • $$88$$
  • $$22$$
  • $$20$$
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