Relations and Functions Test

Result

Relations and Functions Test - 60

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

If $$f:[0, \infty)\to [0,\infty)$$ and $$f(x) = \dfrac{x}{1+x}$$, then $$f$$ is

A
One-one and onto

B
One-one but not onto

C
Onto but not one-one

D
Neither one-one nor onto

Solution

Here, $$f:[0, \infty) \rightarrow [0, \infty)$$ i.e, domain is $$[0, \infty)$$ and codomain is $$[0, \infty)$$
For one-one
$$f(x) = \dfrac{x}{1+x}$$
$$f'(x) = \dfrac{1}{(1+x)^2} > 0, \forall x \in [0, \infty)$$
$$\therefore f(x)$$ is increasing in this domain. Thus $$f(x)$$ is one-one in its domain
For onto (we find range)
$$f(x) = \dfrac{x}{1+x}$$ i.e. $$y =\dfrac{x}{1+x}$$
$$\Rightarrow y+yx=x$$
$$\Rightarrow x=\dfrac{y}{1-y}$$
$$\Rightarrow x=\dfrac{y}{1-y} \ge 0$$ as $$x \ge 0$$
$$\therefore 0 \le y \neq 1$$ and $$y<1$$
$$\therefore$$ Range $$\neq$$ Codomain
$$\therefore f(x)$$ is one-one but not onto
Question 2
1 / -0

If $$\ast $$ is the operation defined by $$a\ast b={ a }^{ b }$$ for $$a,b\in N$$, then $$\left( 2\ast 3 \right) \ast 2$$ is equal to

A
$$81$$

B
$$512$$

C
$$216$$

D
$$64$$

E
$$243$$

Solution

Given $$a\ast b={ a }^{ b }$$
$$\therefore (2\ast 3)\ast 2={ (2 }^{ 3 })\ast 2=64$$
Question 3
1 / -0

The function $$f:A\rightarrow B$$ given by $$f(x) = x ,x\in A$$, is one to one but not onto. Then;

A
$$B\subset A$$

B
$$A=B$$

C
$$A'\subset B'\quad $$

D
$$A\subset B$$

E
$$A'\cap B'=\phi $$

Solution

$$f(x)=x$$ is identity function
It is given that $$f(x)$$ is one to one but not on to.
It means some element in $$B$$ has no pre image in $$A$$
$$A\subset B$$
Question 4
1 / -0

If $$fog = |\sin x|$$ and $$gof = \sin^{2}\sqrt {x}$$, then $$f(x)$$ and $$g(x)$$ are

A
$$f(x) = \sqrt {\sin x}, g(x) = x^{2}$$

B
$$f(x) = |x|, g(x) = \sin x$$

C
$$f(x) = \sqrt {x}, g(x) = \sin^{2}x$$

D
$$f(x) = \sin \sqrt {x}, g(x) = x^{2}$$

Solution

$$fog = f\left \{g(x)\right \} = |\sin x| = \sqrt {\sin^{2}x}$$

Also, $$gof = g\left \{f(x)\right \} = \sin^{2}\sqrt {x}$$

Obviously, $$\sqrt {\sin^{2}x} = \sqrt {g(x)}$$

and $$\sin^{2} \sqrt {x} = \sin^{2} \left \{f(x)\right \}$$
i.e., $$g (x) = \sin^{2}x$$

and $$f(x) = \sqrt {x}$$.
Question 5
1 / -0

Let $$f(x) = |x - 2|$$, where $$x$$ is a real number. Which one of the following is true?

A
$$f$$ is periodic

B
$$f(x + y) = f(x) + f(y)$$

C
$$f$$ is an odd function

D
$$f$$ is not one-one function

E
$$f$$ is an even function

Solution

From the graph of the function it can be observed clearly that $$f$$ is not one-one function.
So option $$D$$ is correct.
Question 6
1 / -0

Inverse of function $$f(x) = \dfrac {10^{x} - 10^{-x}}{10^{x} + 10^{-x}}$$ is

A
$$\log_{10}(2 - x)$$

B
$$\dfrac {1}{2}\log_{10}\left (\dfrac {1 + x}{1 - x}\right )$$

C
$$\dfrac {1}{2}\log_{10}(2x - 1)$$

D
$$\dfrac {1}{2}\log_{10}\left (\dfrac {2x}{2 - x}\right )$$

Solution

Let $$f(x) = y$$, then

$$\dfrac {10^{x} - 10^{-x}}{10^{x} + 10^{-x}} = y$$

$$\Rightarrow \dfrac {10^{2x} - 1}{10^{2x} + 1} = y$$

$$\Rightarrow 10^{2x} = \dfrac {1 + y}{1 - y}$$ .... By Componendo and dividendo

$$\Rightarrow x = \dfrac {1}{2}\log_{10}\left (\dfrac {1 + y}{1 - y}\right )$$

$$\Rightarrow f^{-1}(y) = \dfrac {1}{2}\log_{10}\left (\dfrac {1 + y}{1 - y}\right )$$

$$\therefore f^{-1} (x) = \dfrac {1}{2}\log_{10}\left (\dfrac {1 + x}{1 - x}\right )$$.
Question 7
1 / -0

The value of $$\alpha (\neq 0)$$ for which the function $$f(x) = 1 + \alpha x$$ is the inverse of itself is

A
$$-2$$

B
$$2$$

C
$$-1$$

D
$$1$$

Solution

We have, $$f(x)=1+\alpha x$$
or, $$y=1+\alpha x$$
or, $$x=\cfrac{y-1}{\alpha}$$
Change $$x$$ with $$y$$ and $$y$$ with $$x$$
or, $$y=\cfrac{x-1}{\alpha}$$
or, $$f^{-1}(x)=\cfrac{x-1}{\alpha}$$
For, $$f(x)=f^{-1}(x)$$ value of $$\alpha$$ should be $$-1.$$
Hence, C is the correct option.
Question 8
1 / -0

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

A
1340

B
1860

C
1430

D
1880

E
1680

Solution

Given, $$A = \{1, 3, 5, 7\}$$
and $$B =\{1, 2, 3, 4, 5, 6, 7, 8\}$$
Here, $$n(A) = 4$$ and $$n(B) = 8$$
Thus number of one-one function from $$A$$ into $$B$$
$$= \,^8 P_4 = 8.7.6.5 = 1680$$
Question 9
1 / -0

If $$f(x)=3x+5$$ and $$g(x)={ x }^{ 2 }-1$$, then $$\left( f\circ g \right) $$ $$({ x }^{ 2 }-1)$$ is equal to

A
$$3{ x }^{ 4 }-3x+5$$

B
$$3{ x }^{ 4 }-6{ x }^{ 2 }+5$$

C
$$6{ x }^{ 4 }+3{ x }^{ 2 }+5$$

D
$$6{ x }^{ 4 }-6x+5$$

E
$$3{ x }^{ 2 }+6x+4$$

Solution

$$f(x)=3x+5$$
and $$g(x)={x}^{2}-1$$
$$\therefore f\circ g\left( { x }^{ 2 }-1 \right) =f[g\left( { x }^{ 2 }-1 \right) ]=f\left[ { \left( { x }^{ 2 }-1 \right) }^{ 2 }-1 \right] $$
$$=3\left( { x }^{ 4 }-2{ x }^{ 2 } \right) +5=3{ x }^{ 4 }-6{ x }^{ 2 }+5$$
Question 10
1 / -0

If $$g(x)=1+\sqrt{x}$$ and $$f\{g(x)\}=3+2\sqrt{x}+x$$, then $$f(x)$$ is equal to

A
$$1+2x^2$$

B
$$2+x^2$$

C
$$1+x$$

D
$$2+x$$

Solution

Given, $$g(x)=1+\sqrt{x}$$
and $$f\left \{ g(x) \right \}=3+2\sqrt{x}+x$$ ..... $$(i)$$
$$\Rightarrow f(1+\sqrt{x})=3+2\sqrt{x}+x$$
Put $$1+\sqrt{x}=y\Rightarrow x=(y-1)^2$$
$$\therefore f(y)=3+2(y-1)+(y-1)^2=3+2y-2+y^{2}-2y+1=2+y^2$$
$$\therefore f(x)=2+x^2$$

User

Question Analysis

Correct -
Wrong -
Skipped -

My Perfomance

Score
-
out of -
Rank
-
out of -

Re-Attempt Weekly Quiz Competition

Relations and Functions Test - 60

Result

Relations and Functions Test - 60

Score

-

Rank

-

SHARING IS CARING

Question 1

One-one and onto

One-one but not onto

Onto but not one-one

Neither one-one nor onto

Solution

Question 2

$$81$$

$$512$$

$$216$$

$$64$$

$$243$$

Solution

Question 3

$$B\subset A$$

$$A=B$$

$$A'\subset B'\quad $$

$$A\subset B$$

$$A'\cap B'=\phi $$

Solution

Question 4

$$f(x) = \sqrt {\sin x}, g(x) = x^{2}$$

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

$$f(x) = \sqrt {x}, g(x) = \sin^{2}x$$

$$f(x) = \sin \sqrt {x}, g(x) = x^{2}$$

Solution

Question 5

$$f$$ 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

Solution

Question 6

$$\log_{10}(2 - x)$$

$$\dfrac {1}{2}\log_{10}\left (\dfrac {1 + x}{1 - x}\right )$$

$$\dfrac {1}{2}\log_{10}(2x - 1)$$

$$\dfrac {1}{2}\log_{10}\left (\dfrac {2x}{2 - x}\right )$$

Solution

Question 7

$$-2$$

$$2$$

$$-1$$

$$1$$

Solution

Question 8

1340

1860

1430

1880

1680

Solution

Question 9

$$3{ x }^{ 4 }-3x+5$$

$$3{ x }^{ 4 }-6{ x }^{ 2 }+5$$

$$6{ x }^{ 4 }+3{ x }^{ 2 }+5$$

$$6{ x }^{ 4 }-6x+5$$

$$3{ x }^{ 2 }+6x+4$$

Solution

Question 10

$$1+2x^2$$

$$2+x^2$$

$$1+x$$

$$2+x$$

Solution

User

Question Analysis

My Perfomance

Score

-

Rank

-