CBSE Questions for Class 11 Commerce Applied Mathematics Functions Quiz 6 - MCQExams.com

Let $$f : R \rightarrow R$$ be defined by $$f(x) = x^4$$ then
  • f may be one-one and onto
  • f is one-one and onto
  • f is one-one but not onto
  • f is neither one-one nor onto
If $$f:[0, \infty)\to [0,\infty)$$ and $$f(x) = \dfrac{x}{1+x}$$, then $$f$$ is 
  • One-one and onto
  • One-one but not onto
  • Onto but not one-one
  • Neither one-one nor onto
The function $$f:A\rightarrow B$$ given by $$f(x) = x ,x\in A$$, is one to one but not onto. Then;
  • $$B\subset A$$
  • $$A=B$$
  • $$A'\subset B'\quad $$
  • $$A\subset B$$
  • $$A'\cap B'=\phi $$
The domain of definition of function $$f(x) = \dfrac {1 + 2(x + 4)^{-0.5}}{2 - (x + 4)^{0.5}} + (x + 4)^{0.5} + 4(x + 4)^{0.5}$$ is
  • $$R$$
  • $$(-4, 4)$$
  • $$R^{+}$$
  • $$(-4, 0)\cup (0, \infty)$$
If $$fog = |\sin x|$$ and $$gof = \sin^{2}\sqrt {x}$$, then $$f(x)$$ and $$g(x)$$ are
  • $$f(x) = \sqrt {\sin x}, g(x) = x^{2}$$
  • $$f(x) = |x|, g(x) = \sin x$$
  • $$f(x) = \sqrt {x}, g(x) = \sin^{2}x$$
  • $$f(x) = \sin \sqrt {x}, g(x) = x^{2}$$
The domain of the function $$ f(x) = \sin^{-1} \left( \dfrac {x+5}{2} \right) $$ is :
  • $$ [-1.1]$$
  • $$[2,3]$$
  • $$[3,7]$$
  • $$[-7,-3]$$
  • $$(- \infty , \infty )$$
Let $$f(x) = |x - 2|$$, where $$x$$ is a real number. Which one of the following is true?
  • $$f$$ is periodic
  • $$f(x + y) = f(x) + f(y)$$
  • $$f$$ is an odd function
  • $$f$$ is not one-one function
  • $$f$$ is an even function
If $$A = \left \{ 1 , 3 , 5 , 7 \right \} $$ and $$ B = \left \{ 1 , 2 , 3, 4 , 5 , 6 , 7 , 8 \right \} $$ then the number of one-to-one functions from $$A$$ into $$B$$ is 
  • 1340
  • 1860
  • 1430
  • 1880
  • 1680
If $$f(x)=3x+5$$ and $$g(x)={ x }^{ 2 }-1$$, then $$\left( f\circ g \right) $$ $$({ x }^{ 2 }-1)$$ is equal to
  • $$3{ x }^{ 4 }-3x+5$$
  • $$3{ x }^{ 4 }-6{ x }^{ 2 }+5$$
  • $$6{ x }^{ 4 }+3{ x }^{ 2 }+5$$
  • $$6{ x }^{ 4 }-6x+5$$
  • $$3{ x }^{ 2 }+6x+4$$
If $$g(x)=1+\sqrt{x}$$ and $$f\{g(x)\}=3+2\sqrt{x}+x$$, then $$f(x)$$ is equal to
  • $$1+2x^2$$
  • $$2+x^2$$
  • $$1+x$$
  • $$2+x$$
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 $$f : R - \left \{-1, k\right \} \rightarrow R - \left \{\alpha, \beta \right \}$$ is a bijective function defined by $$f(x) = \dfrac {(2x - 1)(2x^{2} - 4px + p^{3})}{(x + 1)(x^{2} - p^{2}x + p^{2})}$$ (where $$p\geq 0$$), then identify which of the following statement(s) is (are) correct?
  • If $$k\epsilon (-1, 1)$$ then $$\alpha + \beta = 2$$
  • If $$k\epsilon (1, 3)$$ then $$\alpha + \beta = 6$$
  • If $$k\epsilon (1, 3)$$ then $$\alpha + \beta = 4$$
  • If $$k\epsilon (-1, 1)$$ then $$\alpha + \beta = 6$$
If $$f(x)=\left| x \right| ,x\in R$$, then
  • $$f(x)=\left( f\times f \right) \left( x \right) $$
  • $$f(x)=x$$
  • $$f(x)=\left( f\times f \right) \left( x^2 \right) $$
  • $$f(x)=\left( f\circ f \right) \left( x \right) $$
If $$g(x)={ x }^{ 2 }+x-2$$ and $$\cfrac { 1 }{ 2 } (g\circ f)(x)=2{ x }^{ 2 }-5x+2$$, then $$f(x)$$ is
  • $$2x-3$$
  • $$2x+3$$
  • $$2{ x }^{ 2 }+3x+1$$
  • $$2{ x }^{ 2 }+3x-1$$
If $$(ax^2 + bx + c)y +a'x^2+b'x+c=0$$, then the condition that x may be a rational function of y is
  • $$(ac'-a'c)^2=(ab'-a'b)(bc'-b'c)$$
  • $$(ab'-a'b)^2=(ab'-a' c)(bc'-b'c)$$
  • $$(bc'-b'c)^2=(ab'-a'b)(ac'-a'c)$$
  • None of these
If $$f(x)=\sin ^{ 2 }{ x } +\sin ^{ 2 }{ \left( x+\cfrac { \pi  }{ 3 }  \right)  } +\cos { x } \cos { \left( x+\cfrac { \pi  }{ 3 }  \right)  } $$ and $$g\left( \cfrac { 5 }{ 4 }  \right) =1$$, then $$g\circ f(x)$$ is equal to
  • $$0$$
  • $$1$$
  • $$\sin { { 1 }^{ o } } $$
  • None of these
If $$f(x)=ax+b $$ and $$g(x)=cx+d$$, then $$f\left( g(x) \right) =g\left( f(x) \right) \Leftrightarrow$$
  • $$f(a)=g(c)$$
  • $$f(b)=g(b)$$
  • $$f(d)=g(b)$$
  • $$f(x)=g(a)$$
Number of solution of the equation  $$f ( x ) = g ( x )$$  are same as number of point of intersection of the curves $$y = f ( x )$$  and  $$y = g ( x )$$  hence answer the following question.
Number of the solution of the equation  $$2 ^ { x } = | x - 1 | + | x + 1 |$$  is
  • $$0$$
  • $$1$$
  • $$2$$
  • $$\infty$$
The domain of definition of the function y(x) given by the equation $$a^x + a^y = a(a>1)$$ is
  • $$0 < x \leq 1$$
  • $$0$$ $$\leq$$ $$x < 1$$
  • $$- \infty < x < 1$$
  • $$- \infty < x \leq 0$$.
Let $$f(x)={ x }^{ 3 }-3x+1$$. The number of different real solutions of $$f(f(x))=0$$
  • $$2$$
  • $$4$$
  • $$5$$
  • $$7$$
The domain of the function $$f(x)=\cos ^{ -1 }{ \left( \cfrac { 1-\left| x \right|  }{ 2 }  \right)  } $$ is
  • $$\left( -\infty ,-3 \right) \cup \left( 3,\infty \right) $$
  • $$\left[ -3,3 \right] $$
  • $$\quad (-\infty ,-3]\cup [3,\infty )$$
  • $$\phi $$
 If $$f:R\rightarrow R$$, $$g:R\rightarrow R$$ are defined by$$ f(x)=5x-3$$, $$g(x)=x^{2}+3$$, then $$(gof^{-1})(3)$$=
  • $$\dfrac{25}{3}$$
  • $$ \dfrac{111}{25}$$
  • $$\dfrac{9}{25} $$
  • $$\dfrac{25}{111} $$
If the real-valued function $$f(x)=px+\sin x$$ is a bijective function, then the set of all possible values of $$p\epsilon R$$ is?
  • $$R-\{0\}$$
  • $$R$$
  • $$(0, \infty)$$
  • None of these
$$f,g:R\rightarrow R$$ 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
  • $$\cfrac { 2 }{ 30+x } $$
  • $$\cfrac { 2 }{ 30-x } $$
  • $$\cfrac { 2 }{ 3\left( 28-\pi \right) } $$
  • $$\cfrac { 2 }{ 3\left( 28+\pi \right) } $$
The graph of a constant function $$f(x)=k$$ is?
  • A straight line parallel to $$X$$-axis
  • A straight line parallel to $$Y$$-axis
  • A straight line passing through orgin
  • None
If $$f,g,h$$ are three functions from a set of positive real numbers into itself satisfying the condition,
$$f(x) \cdot g(x)=h \sqrt{x^2 + y^2}$$ such that $$x,y \epsilon (0,\infty)$$.then, $$\dfrac{f(x)}{g(x)}$$ is a?
  • Constant function
  • Identity function
  • Zero function
  • Signum function
A constant function is a periodic function.
  • True
  • False
If $$f: R\rightarrow R$$ and $$g: R\rightarrow R$$ are defined $$f(x) = x - [x]$$ and $$g(x) = [x]\forall x\epsilon R, f(g(x))$$.
  • $$x$$
  • $$0$$
  • $$f(x)$$
  • $$g(x)$$
Find number of all such function $$y = f(x)$$ which are onto?
  • $$243$$
  • $$93$$
  • $$150$$
  • None of these
On differentiating an identity function, we get?
  • Signum function
  • Sinc function
  • Constant function
  • None
Find the domain of definition of $$f(x) = \dfrac {\log_{2}(x + 3)}{x^{2} + 3x + 2}$$.
  • $$(-3, \infty)$$
  • $$\left \{-1, -2\right \}$$
  • $$(-3, \infty) - \left \{-1, -2\right \}$$
  • $$(-\infty, \infty)$$
If $$f(x_{1})=f(x_{2})\Rightarrow x_{1}=x_{2}\forall x_{1},x_{2}\epsilon A$$, then what type of a function is $$f:A\rightarrow B?$$
  • One-one
  • Constant
  • Onto
  • Many one
If $$g(x)-x^{2}-x+1 and f(x)=\sqrt{\frac{1}{x}-x}$$, then-
  • Domain of f(g(x)) ids[0,1]
  • Range of f(g(x)) is $$(0,\frac{7}{2\sqrt{3}})$$
  • f (g(x)) is many -one function
  • f(g(x)) is unbounded function
The domain of the function $$f\left( x \right) =\dfrac { 1 }{ \sqrt { \left| x \right| -x }  }$$ is
  • $$\left( 0,\infty \right) $$
  • $$\left( -\infty ,0 \right) $$
  • $$\left( -\infty ,\infty \right) -\left\{ 0 \right\} $$
  • $$\left( -\infty ,\infty \right) $$
If A = {1, 3, 5, 7} , B = {2, 4, 6, 8, 10} and let R = {(1,8), (3,6), (5,2), (1,4)} be a relation from A to B. Then,
Domain (R) = ?
  • $$\{1, 3, 5\}$$
  • $$\{8, 6, 2, 4\}$$
  • $$\{1, 2, 3, 4\}$$
  • None of these
Consider the following functions are odd function in their default domains
(i) $$\cfrac { { 2 }^{ x }-1 }{ { 2 }^{ x }+1 } $$
(ii) $$\cfrac { { x }^{ 2 }+1 }{ x\sin { x }  } $$
(iii) $$\ln { \left( \cfrac { 1+x }{ 1-x }  \right)  } $$
(iv) $$x{ e }^{ \left| x \right| +\cos { x }  }\quad $$
Which of these is/are odd
  • (i) and (iii)
  • (i) and (iv)
  • all four
  • (i), (iii) and (iv)
Let $$f$$ be an injective map with domain {x, y, z} and range {1, 2, 3} such that exactly one of the following statements is correct and the remaining are false :
$$f (x) = 1, f (y) \sqrt 1, f (z) \sqrt 2$$. The value of $$f^{-1} (1)$$ is
  • x
  • y
  • z
  • none of these
Let $$f(x)=\dfrac{[x]} {[x+2]}$$. Find the domain of $$f(x)$$
  • $$x\,\epsilon R, x$$ is not an integer
  • $$(-\infty,-2)\cup[-1,\infty)$$
  • $$x\,\epsilon R, x\neq 1$$
  • $$(-\infty,-1]$$
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, $$(gof^{-1})(3) =$$
  • $$\displaystyle \frac{25}{3}$$
  • $$\displaystyle \frac{111}{25}$$
  • $$\displaystyle \frac{9}{25}$$
  • $$\displaystyle \frac{25}{111}$$

A function R on the set N of natural numbers is defined as R ={$$(2n, 2n+1):n\in N$$} 
The domain of R= {2, 4, 6, 8,............}
  • True
  • False
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:
  • $$\pi$$
  • $$0$$
  • $$1$$
  • $$-1$$
$$f: ( 0,\infty )\rightarrow [0,\infty )$$  defined by $$f(x)=x^{2}$$  is 
  • one - one but not onto
  • onto but not one - one
  • bijective
  • neither one - one nor onto
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 $$2$$
Let $$A = \{ 1,2,3,4,5,6\} .$$ The number of onto functions from $$A$$ to$$A$$ such that.$$f\left( x \right) \ne x$$ for all $$x \in A$$ is
  • $$720$$
  • $$240$$
  • $$245$$
  • $$265$$
The domain of definition of the function y(x) given by equation $${2^x} + {2^y} = 2$$ is
  • $$0 < x \le 1$$
  • $$0 \le x \le 1$$
  • $$ - \infty < x \le 0$$
  • $$ - \infty < x < 1$$
The domain of the function $$f(x) = sin^{-1} (log_2(\dfrac{x^2}{2})) $$ is 
  • $$[ -2, -1) \cup (1, 2]$$
  • $$( -2, -1] \cup [1, 2]$$
  • $$[ -2, -1] \cup [1, 2]$$
  • $$( -2, -1) \cup (1, 2)$$
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
  • $$2x-3$$
  • $$2x+3$$
  • $$2{x^2} + 3x + 1$$
  • $$2{x^2} - 3x -1$$
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