Relations and Functions Test

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Relations and Functions Test - 62

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Weekly Quiz Competition

Question 1
1 / -0

$$f,g:R\rightarrow R$$ 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

A
$$\cfrac { 2 }{ 30+x } $$

B
$$\cfrac { 2 }{ 30-x } $$

C
$$\cfrac { 2 }{ 3\left( 28-\pi \right) } $$

D
$$\cfrac { 2 }{ 3\left( 28+\pi \right) } $$

Solution

Given that,
$$g:R \to R$$

$$f\left( x \right) = 3x - \sin \left( {\dfrac{{\pi x}}{2}} \right)$$

$$g\left( x \right) = {x^3} + 2x - \sin \left( {\dfrac{{\pi x}}{2}} \right)$$

$$\therefore \dfrac{{d{f^{ - 1}}}}{{dx}}{\left( {g'\left( x \right)} \right)_{x = 12}} = \dfrac{2}{{30 + x}}$$

Hence, the option $$(A)$$ is correct.
Question 2
1 / -0

If $$f: R\rightarrow R$$ and $$g: R\rightarrow R$$ are defined $$f(x) = x - [x]$$ and $$g(x) = [x]\forall x\epsilon R, f(g(x))$$.

A
$$x$$

B
$$0$$

C
$$f(x)$$

D
$$g(x)$$

Solution

We have, $$f(x)=x-[x]=\{x\}\dots (1)$$

where $$\{x\}$$ is the fractional part of $$x$$

If $$g(x)=[x]$$, where $$[x]$$ is the greatest integer part of $$x$$, then

The value of $$g(x)$$ will always be an integer.

And $$\therefore f(g(x))=f([x])=0$$

$$\because $$ the fractional part of $$g(x)$$ i.e $$[x]$$ is always $$0$$.
Question 3
1 / -0

If $$f: (-\infty, 3]\rightarrow [7, \infty); f(x) = x^{2} - 6x + 16$$, then which of the following is true?

A
$$f^{-1}(x) = 3 + \sqrt {x - 7}$$

B
$$f^{-1}(x) = 3 - \sqrt {x - 7}$$

C
$$f^{-1}(x) = \dfrac {1}{x^{2} - 6x + 16}$$

D
$$f$$ is many-one

Solution

Let $$f(x) = y $$

$$=x^2 - 6x + 16$$

$$= (x-3)^2+7$$

So $$(x-3)^2 = y -7$$

$$\Rightarrow x = 3+ \sqrt{y-7}$$

now $$f(x) = y $$

$$\Rightarrow x = f^{-1}(y)$$

So $$f^{-1}(y) = 3 + \sqrt{y-7}$$

Since x,y are arbitrary variables replace y with x to get Option A as the answer.
Question 4
1 / -0

Let $$f(x)={ x }^{ 3 }-3x+1$$. The number of different real solutions of $$f(f(x))=0$$

A
$$2$$

B
$$4$$

C
$$5$$

D
$$7$$
Question 5
1 / -0

The inverse of the function $$y = 5^{\ln\ x}$$ is

A
$$x = y^{\frac {1}{\ln 5}}, y > 0$$

B
$$x = y^{\ln 5}, y > 0$$

C
$$x = y^{\frac {1}{\ln 5}}, y < 0$$

D
$$x = 5\ln y, y > 0$$

Solution

Given that
$$ y = 5^{\ln x} $$

$$ \ln y = \ln (5^{\ln x}) $$
$$ = (\ln x) (\ln 5)$$

$$ \therefore \ln x = \dfrac{\ln y}{\ln 5} $$

$$ x = e^{\frac{\ln y}{\ln 5} }$$
$$ = (e^{\ln y})^{\frac{1}{\ln 5}}$$
$$ = y^{\frac{1}{\ln 5}}$$

Since the domain of the natural log function ln() is the set of positive real numbers, it is required here that
$$ y > 0$$.

The inverse function:
$$ x = y^{\frac{1}{\ln 5}}, \; \; y > 0$$
Question 6
1 / -0

If $$f(x)$$ is a real valued function, then which of the following is one-one function?

A
$$f(x)=e^{|x|}$$

B
$$f(x)=|e^x|$$

C
$$f(x)=\sin x$$

D
$$f(x)=|\sin x|$$

Solution

$$(1)$$ $$f(x)=e^{|x|}$$
Since$$f(x)=f(-x)$$
So $$f(x)$$ is many one.

$$(2)$$ $$f(x)=|e^{x}|$$
so, $$f(x)=e^x$$
and as $$e^x$$ is one-one
So $$f(x)$$ is one-one.

$$(3)$$ $$f(x)=\sin x$$
Since $$\sin x$$ is many one
So $$f(x)$$ is also many-one.

$$(4)$$ $$f(x)=|\sin x|$$
Since $$\sin x$$ is many one
So $$f(x)$$ is also many-one.
Question 7
1 / -0

If $$A =\{1, 2, 3\}$$ and $$ B = \{4, 5\}$$ then the number of function $$f : A \rightarrow B$$ which is not onto is ______

A
$$2$$

B
$$6$$

C
$$8$$

D
$$4$$

Solution

Total number of function $$f:A\rightarrow B$$ is the number of relations from $$A$$ to $$B$$ such that all the elements of $$A$$ are in the first elements of function $$f$$.
Hence, total number of functions are $$=2\times 2\times 2 =2^3=8$$
There are two functions for which the function is $$\{(1,4),(2,4),(3,4)\}$$ and $$\{(1,5),(2,5),(3,5)\}$$.
Hence there are two non-onto functions.
Question 8
1 / -0

If $$f\,: R \rightarrow R, g: R \rightarrow R\,$$ are defined by $$f(x)= 5x -3,g(x)=x^2 + 3$$, then, $$(gof^{-1})(3) =$$

A
$$\displaystyle \frac{25}{3}$$

B
$$\displaystyle \frac{111}{25}$$

C
$$\displaystyle \frac{9}{25}$$

D
$$\displaystyle \frac{25}{111}$$

Solution

$$f(x)=5x-3,$$ $$g(x)=x^2+3$$
Let $$f(x)=y$$
$$\Rightarrow$$ $$5x-3=y$$
$$\Rightarrow$$ $$x=\dfrac{y+3}{5}$$ ----( 1 )
Now,
$$f(x)=y$$
$$f^{-1}(y)=x$$
$$f^{-1}(y)=\dfrac{y+3}{5}$$ [ From ( 1 ) ]
$$f^{-1}(x)=\dfrac{x+3}{5}$$
Next,
$$(gof^{-1})(3)$$
$$\Rightarrow$$ $$g[f^{-1}(3)]$$
$$\Rightarrow$$ $$g\left[\dfrac{3+3}{5}\right]$$

$$\Rightarrow$$ $$g\left[\dfrac{6}{5}\right]$$

$$\Rightarrow$$ $$\left(\dfrac{6}{5}\right)^2+3$$ [ Since, $$g(x)=x^2+3$$ ]

$$\Rightarrow$$ $$\dfrac{36}{25}+3$$

$$\Rightarrow$$ $$\dfrac{36+75}{25}$$

$$\Rightarrow$$ $$\dfrac{111}{25}$$
Question 9
1 / -0

Let $$f:A \to b$$ be a function defined by f(x) =$$\sqrt {1 - {x^2}} $$

A
f(x) is one-one if A =[0,1]

B
f(x) is onto if B = [0,1]

C
f(x) is one-one if A =[-1 , 0]

D
f(x) is onto if B = [-1,1]

Solution

$$f:A\rightarrow B$$
$$ f(x)=\sqrt { 1-{ x }^{ 2 } } $$
When $$ x=0$$
$$ f(0)=\sqrt { 1-0 } $$
$$ =1$$
$$ f(1)=\sqrt { 1-{ 1 }^{ 2 } } $$
$$ =0$$
$$ f(x)$$ is one - one function when $$A=[0,1]$$
Question 10
1 / -0

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:

A
$$\pi$$

B
$$0$$

C
$$1$$

D
$$-1$$

Solution

Given that $$g(x)=[x]$$

$$\Rightarrow$$ $$g(\pi )=g(3.141....)$$

i.e $$g(\pi )=3$$

$$(f\circ g)(x)=f(g(x))$$

$$\Rightarrow$$ $$(f\circ g)(\pi ))=f(g(\pi ))$$

But $$g(\pi)=3$$

$$\Rightarrow f(3)=1$$ ...... $$[\because f(x)=1 \text{ for } x>0]$$

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Re-Attempt Weekly Quiz Competition

Relations and Functions Test - 62

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Relations and Functions Test - 62

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SHARING IS CARING

Question 1

$$\cfrac { 2 }{ 30+x } $$

$$\cfrac { 2 }{ 30-x } $$

$$\cfrac { 2 }{ 3\left( 28-\pi \right) } $$

$$\cfrac { 2 }{ 3\left( 28+\pi \right) } $$

Solution

Question 2

$$x$$

$$0$$

$$f(x)$$

$$g(x)$$

Solution

Question 3

$$f^{-1}(x) = 3 + \sqrt {x - 7}$$

$$f^{-1}(x) = 3 - \sqrt {x - 7}$$

$$f^{-1}(x) = \dfrac {1}{x^{2} - 6x + 16}$$

$$f$$ is many-one

Solution

Question 4

$$2$$

$$4$$

$$5$$

$$7$$

Question 5

$$x = y^{\frac {1}{\ln 5}}, y > 0$$

$$x = y^{\ln 5}, y > 0$$

$$x = y^{\frac {1}{\ln 5}}, y < 0$$

$$x = 5\ln y, y > 0$$

Solution

Question 6

$$f(x)=e^{|x|}$$

$$f(x)=|e^x|$$

$$f(x)=\sin x$$

$$f(x)=|\sin x|$$

Solution

Question 7

$$2$$

$$6$$

$$8$$

$$4$$

Solution

Question 8

$$\displaystyle \frac{25}{3}$$

$$\displaystyle \frac{111}{25}$$

$$\displaystyle \frac{9}{25}$$

$$\displaystyle \frac{25}{111}$$

Solution

Question 9

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]

Solution

Question 10

$$\pi$$

$$0$$

$$1$$

$$-1$$

Solution

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