Functions Test 29

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Functions Test 29

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Weekly Quiz Competition

Question 1
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 2
1 / -0

Directions For Questions

Given a function $$f : A \rightarrow B$$; where $$A = \left \{1, 2, 3, 4, 5\right \}$$ and $$B = \left \{6, 7, 8\right \}$$.

...view full instructions

Find number of all such function $$y = f(x)$$ which are onto?

A
$$243$$

B
$$93$$

C
$$150$$

D
None of these

Solution

1. For all possible functions from A to B
2. $${ 3 }^{ 5 }= 243$$ (as each value in A can get matched to any one of the elements in B)

Now we remove functions in which all match to one elements only.
3. Number of such elements $$= 1 + 1 + 1= 3$$

(One when all match to 6, one when all match to 7 and one when all match to 8)

For functions in which 2 elements are only matched.
4. $$3[\binom {5} {1} + \binom {5} {2} + \binom {5} {3} + \binom {5} {4}]=90$$

If we select any two numbers from $$6,7$$ and $$8.$$ Possible ways are $$3$$.

Now we distribute these $$5$$ numbers from $$1$$ to $$5$$ in different possible ways between the two numbers.
This is done by $$[\binom {5} {1} + \binom {5} {2} + \binom {5} {3} + \binom {5} {4}]$$.
5. Hence the total onto functions are $$= 243 - ( 3 + 90 ) = 150$$
Question 3
1 / -0

On differentiating an identity function, we get?

A
Signum function

B
Sinc function

C
Constant function

D
None

Solution

Derivative of an Identity function gives a Constant function

$$\dfrac{d(cx)}{dx} = c$$
Where $$c$$ is a constant.
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

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
$$\dfrac{25}{3}$$

B
$$ \dfrac{111}{25}$$

C
$$\dfrac{9}{25} $$

D
$$\dfrac{25}{111} $$

Solution

$$f:R\rightarrow R$$
$$ g:R\longrightarrow R$$
$$ f\left( x \right) =5x-3$$
$$g\left( x \right) ={ x }^{ 2 }+3$$
$$\left(gof^{-1}\right)\left(3\right)$$
$$f\left(x\right)={5}{x}-{3}$$
Let $$f\left(x\right)=y$$ $$\Rightarrow{x}=f^{-1}\left(y\right)$$
$$f\left(x\right)=y={5}x-3$$
$$\Rightarrow x=\dfrac{y+3}{5}$$
$$\Rightarrow f^{-1}\left(y\right)=\dfrac{y+3}{5}$$
$$ f^{-1}\left(x\right)=\dfrac{x+3}{5}$$ and $$g\left(x\right)=x^{2}+{3}$$
$$\left(gof^{-1}\right)\left(3\right)=g\left[f^{-1}\left(3\right)\right]=g\left(\dfrac{3+3}{5}\right)=g\left(\dfrac{6}{5}\right)$$
$$=\left[\left(\dfrac{6}{5}\right)^{2}+3\right]=\dfrac{36}{25}+{3}=\dfrac{111}{25}$$
Hence, $$\dfrac{111}{25}$$ is the correct answer.
Question 6
1 / -0

Let $$f(x)=\dfrac{[x]} {[x+2]}$$. Find the domain of $$f(x)$$

A
$$x\,\epsilon R, x$$ is not an integer

B
$$(-\infty,-2)\cup[-1,\infty)$$

C
$$x\,\epsilon R, x\neq 1$$

D
$$(-\infty,-1]$$

Solution

As from the definition of $$[x]$$ we get,
$$[x]=\begin{cases}.\\.\\ -1 &\mbox{for}& -1\le x\lt 0 \\ 0 &\mbox{for}&\ 0\le x\lt 1\\ 1 &\mbox{for}& \ 1\le x\lt 2\\.\\.\end {cases}$$.
Now the given function $$f(x)$$ will be defined for $$[x+2]\ne 0$$.
And this occurs of $$x \in (-\infty,-2)\cup [-1,\infty)$$.
So option (B) is correct.
Question 7
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 8
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 9
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 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

Functions Test 29

Result

Functions Test 29

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Rank

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SHARING IS CARING

Question 1

$$x$$

$$0$$

$$f(x)$$

$$g(x)$$

Solution

Question 2

$$243$$

$$93$$

$$150$$

None of these

Solution

Question 3

Signum function

Sinc function

Constant function

None

Solution

Question 4

$$2$$

$$4$$

$$5$$

$$7$$

Question 5

$$\dfrac{25}{3}$$

$$ \dfrac{111}{25}$$

$$\dfrac{9}{25} $$

$$\dfrac{25}{111} $$

Solution

Question 6

$$x\,\epsilon R, x$$ is not an integer

$$(-\infty,-2)\cup[-1,\infty)$$

$$x\,\epsilon R, x\neq 1$$

$$(-\infty,-1]$$

Solution

Question 7

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

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

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

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

Solution

Question 8

$$2$$

$$6$$

$$8$$

$$4$$

Solution

Question 9

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

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

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

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

Solution

Question 10

$$\pi$$

$$0$$

$$1$$

$$-1$$

Solution

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