Relations and Functions Test

Result

Relations and Functions Test - 52

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(x) + f(1-x) = 10 then the value of $$\displaystyle f\left ( \frac{1}{10} \right )+f\left ( \frac{2}{10} \right )+.........+f\left ( \frac{9}{10} \right )$$

A
is 45

B
is 50

C
is 90

D
Cannot be determined

Solution

$$ f(\dfrac {1}{10}) + f(\dfrac {2}{10}) + f(\dfrac {3}{10}) + f(\dfrac {4}{10}) + f(\dfrac {5}{10}) + f(\dfrac {6}{10}) + f(\dfrac {7}{10}) + f(\dfrac {8}{10}) + f(\dfrac {9}{10}) $$

$$ = [f(\dfrac {1}{10}) + f(\dfrac {9}{10}) ] + [ f(\dfrac {2}{10}) + + f(\dfrac {8}{10}) ] + [ f(\dfrac {3}{10}) + + f(\dfrac {7}{10}) ] + [ f(\dfrac {4}{10}) + f(\dfrac {6}{10}) ] + f(\dfrac {5}{10}) ] $$

$$ = 10 + 10 + 10 + 10 + 5 = 45 $$
Question 2
1 / -0

If $$f (x) = 2x - 1$$ and $$g (x) = 3x + 2$$, then find $$(fog) (x)$$ :

A
$$2 (3x + 1)$$

B
$$2 ( 3x + 2)$$

C
$$3 (2x + 1 )$$

D
$$3 ( 3x + 1 )$$

Solution

$$fog(x)=2(3x+2)-1=6x+4-1=6x+3=3(2x+1)$$
Question 3
1 / -0

If $$\displaystyle R=R^{-1}$$ then the relation R is ________

A
reflexive

B
symmetric

C
anti-symmetric

D
transitive

Solution

When a relation is equal to its inverse, then the relation is symmetric.
Question 4
1 / -0

If $$\displaystyle f(x)=2x^{3}+7x-5\, and\, g(x)=f^{-1}(x)$$ then g' (4) is equal to

A
$$\displaystyle \frac{1}{13}$$

B
$$\displaystyle \frac{1}{103}$$

C
$$\displaystyle \frac{1}{4}$$

D
non existent

Solution

Given, $$f(x)=2x^3+7x-5$$
$$\therefore f'(x)= 6x ^2+7$$
$$g(x) =f^{-1}(x)\Rightarrow g\{f(x)\}=x$$
Differentiating both sides, w.r.t $$x$$
$$\displaystyle {g}'\left ( f(x) \right )=\frac{1}{f'\left ( x \right )}=\frac{1}{6x^{2}+7}$$
when $$x = 1,f(x) = 4$$
$$\therefore \displaystyle g'\left ( 4 \right )=\frac{1}{13}$$
Question 5
1 / -0

If $$f(x) = -x^2+1, g(x) = -\sqrt[3]{x}$$ then (gofogofogogog) (x) is.

A
an odd function

B
an even function

C
a polynomial function

D
an identity function

Solution

If $$f(x) = -x^2+1$$......

$$f(x) = -x^2+1, g(x) = -\sqrt[3]{x}$$........

$$\Rightarrow$$ $$f$$ is even and $$g$$ is odd

$$\therefore$$ $$gogog$$ is even

$$\Rightarrow$$ $$fogogog$$ is even

$$\Rightarrow$$ $$gofogofogogog$$ is even.
Question 6
1 / -0

If the function $$f(x) = x^3 + e^{x/2}$$ and $$g(x) = f^{-1}(x)$$; then the value of g'(1) is

A
$$2$$

B
$$-2$$

C
$$1$$

D
$$0$$

Solution

$$f(x) =x^3+e^{x/2}$$
$$\displaystyle f'(x) = 3x^2+\frac{1}{2}e^{x/2}$$
Given g is inverse of f $$\Rightarrow g(f(x))=x$$
Differentiating both sides w.r.t $$x$$
$$\displaystyle g'(f(x)).f'(x) =1\Rightarrow g'(f(x))=\frac{1}{f'(x)} $$
Clearly $$\displaystyle f'(0)=1\therefore g'(1)=g'(f(0))=\frac{1}{f'(0)}=2$$
Question 7
1 / -0

If f(x)=2x-1 and g(x)=3x+2 then find (fog) (x)

A
2(3x+1)

B
2(3x+2)

C
3(2x+1)

D
3(3x+1)

Solution

$$(fog)(x) = f(g(x)) = 2(3x+2) -1 = 6x+4-1 = 6x+3 = 3(2x+1) $$
Question 8
1 / -0

If f(x) = 2x+1 and g(x) = 3x-5 then find $$\left ( fog \right )^{-1}\left ( 0 \right )$$

A
5/3

B
3/2

C
2/3

D
3/5

Solution

$$ (fog)(x) = f(g(x)) = 2(3x-5) + 1 = 6x-10 + 1 = 6x-9 $$

Let $$ (fog)(x)= y $$
Then, to find $$ (fog)^{-1} (x)$$, we find $$ x $$ in terms of $$y $$

So, $$ 6x-9 = y $$
$$ => x = \dfrac {y+9}{6} $$

$$ => (fog)^{-1} (x)= \dfrac {y+9}{6} $$

$$ => (fog)^{-1} (0)= \dfrac {0+9}{6} = \dfrac {3}{2} $$
Question 9
1 / -0

If X = {2,3,5,7,11} and Y = {4,6,8,9,10} then find the number of one-one functions from X to Y

A
720

B
120

C
24

D
12

Solution

Both sets X and Y have the same number of elements.

When two sets have same number of $$ n $$ elements, then the number of one to one functions from one set to the other is $$ n! $$

So, number of one to one functions from X to Y $$ = 5 ! = 5 \times 4 \times 3 \times 2 \times 1 = 120 $$
Question 10
1 / -0

Find $$\left( f\circ g \right) \left( 3 \right) $$ when $$f\left( x \right) =7x-6$$ and $$g\left( x \right) =5{ x }^{ 2 }-7x-6$$.

A
$$-36$$

B
$$1014$$

C
$$-90$$

D
$$120$$

Solution

We have, $$f(x)=7x-6$$ and $$g(x)=5x^2-7x-6$$
Then,$$(f\circ g)(x)$$
$$=f(g(x))$$
$$=f(5x^2-7x-6)$$
$$=7(5x^2-7x-6)-6$$
$$=35x^2-49x-42-6$$
$$=35x^2-49x-48$$
$$\therefore (f\circ g)(3)$$
$$=35(3)^2-49(3)-48$$
$$=315-147-48$$
$$=120$$
So, $$\text{D}$$ is the correct option.

User

Question Analysis

Correct -
Wrong -
Skipped -

My Perfomance

Score
-
out of -
Rank
-
out of -

Re-Attempt Weekly Quiz Competition

Relations and Functions Test - 52

Result

Relations and Functions Test - 52

Score

-

Rank

-

SHARING IS CARING

Question 1

is 45

is 50

is 90

Cannot be determined

Solution

Question 2

$$2 (3x + 1)$$

$$2 ( 3x + 2)$$

$$3 (2x + 1 )$$

$$3 ( 3x + 1 )$$

Solution

Question 3

reflexive

symmetric

anti-symmetric

transitive

Solution

Question 4

$$\displaystyle \frac{1}{13}$$

$$\displaystyle \frac{1}{103}$$

$$\displaystyle \frac{1}{4}$$

non existent

Solution

Question 5

an odd function

an even function

a polynomial function

an identity function

Solution

Question 6

$$2$$

$$-2$$

$$1$$

$$0$$

Solution

Question 7

2(3x+1)

2(3x+2)

3(2x+1)

3(3x+1)

Solution

Question 8

5/3

3/2

2/3

3/5

Solution

Question 9

720

120

24

12

Solution

Question 10

$$-36$$

$$1014$$

$$-90$$

$$120$$

Solution

User

Question Analysis

My Perfomance

Score

-

Rank

-

Results Announcement on: coming Monday

Stay Tuned: We’ll update you on your result via email or SMS.