Relations and Functions Test

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Relations and Functions Test - 30

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Weekly Quiz Competition

Question 1
1 / -0

If $$f(x) = 3x + 2, g(x) = x^2 + 1$$, then the value of $$(fog) (x^2 +1)$$ is

A
$$3x^4 + 6x^2 + 8$$

B
$$3x^4 + 3x + 4$$

C
$$6x^4 + 3x^2 + 2$$

D
$$3x^2 + 6x + 2$$

Solution

We have $$f(x) = 3x + 2, g(x) = x^2 + 1$$
$$fog(x) = f[g(x)] =3(x^2+1)+2= 3x^2 + 5$$
so $$(fog) (x^2 +1) = 3(x^2 + 1)^2 + 5 =3(x^4+2x^2+1)+5= 3x^4+ 6x^2 + 8$$
Question 2
1 / -0

If $$f:A\rightarrow B $$ is surjective then

A
no two elements of $$A$$ have the same image in $$B$$

B
every element of $$A$$ has an image in $$B$$

C
every element of $$B$$ has at least one pre-image in $$A$$

D
$$A$$ and $$B$$ are finite non empty sets

Solution

Surjective means onto function.
co domain $$=$$ Range
So every element of $$B$$ has at least one pre-image in $$A$$.
Question 3
1 / -0

If $$f:(0,\infty )\rightarrow (0,\infty )$$ is defined by $$f(x)=x^{2}$$, then $$f^{-1}(x)=$$

A
$$\sqrt{x}$$

B
$$\dfrac{1}{\sqrt{x}}$$

C
Not invertible

D
$$\dfrac{2}{\sqrt{x}}$$

Solution

As $$f$$ is defined from $$\left ( 0,\infty \right )\rightarrow \left ( 0,\infty \right )$$
$$\therefore f$$ is onto and one - one.
Let $$f\left ( x \right )=x^{2}=y$$
$$\Rightarrow x=f^{-1}\left ( y \right )$$
Also, $$x^2=y\Rightarrow x=\sqrt{y}$$ as $$x> 0$$
$$\therefore f^{-1}\left ( y \right )=\sqrt{y}$$
Hence, $$f^{-1}\left ( x \right )=\sqrt{x}$$
Question 4
1 / -0

$$f:R\rightarrow R , g:R\rightarrow R$$ and $$f(x)= \sin x$$, $$g(x)=x^{2}$$ then $$fog(x)=$$

A
$$x^{2}+\sin x$$

B
$$x^{2}\sin x$$

C
$$\sin^{2}x$$

D
$$\sin x^{2}$$

Solution

$$fog\left ( x \right )=f\left(g ( x \right ))=f (x^{2})$$
$$=\sin x^{2}$$
Question 5
1 / -0

Find the value of $$\displaystyle \left( g\circ f \right) \left( 6 \right) $$ if $$\displaystyle g\left( x \right) ={ x }^{ 2 }+\frac { 5 }{ 2 } $$ and $$\displaystyle f\left( x \right) =\frac { x }{ 4 } -1$$.

A
2.75

B
3

C
3.5

D
8.625

Solution

$$f(x)=\dfrac {x}{4}-1$$ and $$g(x)=x^2+\dfrac {5}{2}$$
$$\therefore f(6)=\dfrac{6}{4}-1=\dfrac{3}{2}-1=\dfrac{1}{2}$$
$$\therefore (gof)(6)=g(\dfrac{1}{2})=\left(\dfrac{1}{2}\right)^2+\dfrac{5}{2}=\dfrac{11}{4}=2.75$$
Question 6
1 / -0

The first component of all ordered pairs is called

A
Range

B
Domain

C
Function

D
None of these

Solution

The first components of all order pair is called Domain.
The second components of all ordered pair is called Range.
Question 7
1 / -0

The second component of all ordered pairs of a relation is

A
Range

B
Domain

C
mapping

D
none of these

Solution

The second components of all ordered pairs of a relation is Range.
Question 8
1 / -0

A ______ maps elements of one set to another set.

A
order

B
set

C
relation

D
function

Solution

A relation map elements of one set to another set
$$R:A\to B$$
Elements in A is mapped to elements in set B.
Question 9
1 / -0

Suppose y is equal to the sum of two quantities of which one varies directly as x and the other inversely as x If y = 6 when x = 4 and $$\displaystyle y=\frac{10}{3}$$ when x = 3 then what is the relation between x and y?

A
$$\displaystyle y=x+\frac{4}{x}$$

B
$$\displaystyle y+2x=\frac{4}{x}$$

C
$$\displaystyle y=2x+\frac{8}{x}$$

D
$$\displaystyle y=2x-\frac{8}{z}$$

Solution

(A)If$$y=X+\frac{4}{x}\Rightarrow 4+\frac{4}{4}=3 \neq 6=6$$
(B)$$y+2x=\frac{4}{x}\Rightarrow 6+2(4)=\frac{4}{4}\Rightarrow 6+8\neq 1$$
(C)$$y=2x+\frac{8}{x}\Rightarrow 6=2(4)+\frac{8}{4}\Rightarrow 6\neq 8+2$$
(D)$$y=2x-\frac{8}{x}\Rightarrow 6=2(4)-\frac{8}{4}=8-2=6$$ If x=4 and y=6
$$y=2x-\frac{8}{x}\Rightarrow \frac{10}{3}=2\times 3-\frac{8}{3}\Rightarrow \frac{10}{3}=\frac{10}{3}$$ If x=3 and y=$$y=\frac{10}{3}$$
Then option (D) $$Y=2x-\frac{8}{x}$$
Question 10
1 / -0

If X is brother of the son of Y's son. How is X related to Y?

A
Son

B
Brother

C
Cousin

D
Grandson

E
Uncle

Solution

Son of Y's Son- Grandson, Brother of Y's Grandson- Y's Grandson
Option D is correct.

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Relations and Functions Test - 30

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Relations and Functions Test - 30

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SHARING IS CARING

Question 1

$$3x^4 + 6x^2 + 8$$

$$3x^4 + 3x + 4$$

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

$$3x^2 + 6x + 2$$

Solution

Question 2

no two elements of $$A$$ have the same image in $$B$$

every element of $$A$$ has an image in $$B$$

every element of $$B$$ has at least one pre-image in $$A$$

$$A$$ and $$B$$ are finite non empty sets

Solution

Question 3

$$\sqrt{x}$$

$$\dfrac{1}{\sqrt{x}}$$

Not invertible

$$\dfrac{2}{\sqrt{x}}$$

Solution

Question 4

$$x^{2}+\sin x$$

$$x^{2}\sin x$$

$$\sin^{2}x$$

$$\sin x^{2}$$

Solution

Question 5

2.75

3

3.5

8.625

Solution

Question 6

Range

Domain

Function

None of these

Solution

Question 7

Range

Domain

mapping

none of these

Solution

Question 8

order

set

relation

function

Solution

Question 9

$$\displaystyle y=x+\frac{4}{x}$$

$$\displaystyle y+2x=\frac{4}{x}$$

$$\displaystyle y=2x+\frac{8}{x}$$

$$\displaystyle y=2x-\frac{8}{z}$$

Solution

Question 10

Son

Brother

Cousin

Grandson

Uncle

Solution

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