Functions Test 58

Result

Functions Test 58

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Weekly Quiz Competition

Question 1
1 / -0

State which of the following defines a mapping from A to B, if $$A={a,b,c,}$$ and $$B={x,y,z}.$$

A

B

C

D
None of these.
Question 2
1 / -0

If $$f ( x ) = \sqrt { x ^ { 2 } + 1 } , g ( x ) = \dfrac { x + 1 } { x ^ { 2 } + 1 }$$ and $$h ( x ) = 2 x - 3 ,$$ then $$f ^ { \prime } \left( h ^ { \prime } \left( g ^ { \prime } ( x ) \right) =\right.$$

A
$$0$$

B
$$\dfrac { 1 } { \sqrt { x ^ { 2 } + 1 } }$$

C
$$\dfrac { 2 } { \sqrt { 5 } }$$

D
$$\dfrac { x } { \sqrt { x ^ { 2 } + 1 } }$$

Solution
Question 3
1 / -0

$${ f }:{ R }\rightarrow { R }$$ where $$f ( x ) = \dfrac { x ^ { 2 } + a x + 1 } { x ^ { 2 } + x + 1 }.$$ Complete set of values of $$'a'$$ such that $$f ( x )$$ is onto to is :

A
$$( - \infty , \infty )$$

B
$$( - \infty , 0 )$$

C
$$( 0 , \infty )$$

D
None

Solution

$$f(x)=\dfrac{x^{2}+a x+1}{x^{2}+x+1}$$
$$f: R \rightarrow R$$
domain co-domain
$$D\left(x^{2}+x+1\right)=-3<0$$
Hence $$x^{2}+x+1$$ is always positive
$$x^{2}+a x+1$$ coefficient of $$x^{2}=1>0$$
$$\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow \infty} \dfrac{x^{2}+a x+1}{x^{2}+x+1} \dfrac{\infty}{\infty}$$ form
$$=\lim _{x \rightarrow 0} \dfrac{x^{2}\left(1+\dfrac{a}{x}+\dfrac{1}{x^{2}}\right)}{x^{2}\left(1+\frac{1}{x}+\dfrac{1}{x^{2}}\right)}=\dfrac{1+0+0}{1+0+0}=1$$
similarly
$$lim_{x\rightarrow-\infty} f(x)=1$$
Hence $$f(x)$$ is bounded
Range $$\neq R$$
$$\neq$$ co-domain
$$f(x)$$ cannot be onto for any value of $${a}$$
option $$D$$ is correct.
Question 4
1 / -0

The domain of $$f(x) = \sin^{-1} log_2 (x^2/2)$$ is

A
$$(0, \infty)$$

B
$$(0, \surd{2})$$

C
$$(-1, 0) \cup (0, 1)$$

D
$$[-2, -1] \cup [1, 2]$$

Solution
Question 5
1 / -0

The number of integral values of $$x$$ in the domain of function $$f$$ defined as $$f(x)=\sqrt { \ln { | } \ln { | } x|| } +\sqrt { 7|x|-|x|^{ { 2 } }-10 } $$ is :

A
$$5$$

B
$$6$$

C
$$7$$

D
$$8$$

Solution
Question 6
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The domain and the range of $$f(x) = \cos^{-1} \sqrt {\log_{[x]} \left (\dfrac {|x|}{x}\right )}$$, where $$[\cdot ]$$ denotes the greatest integer function, are respectively.

A
$$[1, \infty), [0, \pi/2]$$

B
$$[2, \infty), [0, \pi/2]$$

C
$$[2, \infty), \left \{\pi/2\right \}$$

D
$$[1, \infty), \left \{0\right \}$$

Solution
Question 7
1 / -0

Choose correct answer (s) from given choice
If f(x) = x + 4, g (x) = 5x and h(x) = 12/x. Find the value of $${ f }^{ -1 }(g(h(6)))$$

A
10

B
14

C
6

D
0
Question 8
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If f(x)=x+tanx and g(x) is inverse of f(x) then g'(x) is equal to

A
$$\dfrac { 1 }{ 1+(g(x)-{ x) }^{ 2 } } $$

B
$$\dfrac { 1 }{ 1-(g(x)-{ x) }^{ 2 } } $$

C
$$\dfrac { 1 }{ 1+(g(x)-{ x) }^{ 2 } } $$

D
$$\dfrac { 1 }{ 2-(g(x)-{ x) }^{ 2 } } $$
Question 9
1 / -0

The domain of $$f(x)=\sqrt { -x^2 }$$ is

A
($$0,\infty $$)

B
($$-\infty , 0$$)

C
($$0$$)

D
($$1,\infty $$)
Question 10
1 / -0

For the function $$F(x)=\sqrt { { 4-x }^{ 2 } } +\sqrt { { x }^{ 2 }-1 } $$

A
Domain is (-2,2)

B
Domain is (-2, -1) (1,2)

C
Range is $$\left( \sqrt { 3 } ,\sqrt { 5 } \right) $$

D
Range is $$\left( \sqrt { 3 } ,\sqrt { 6 } \right) $$