Relations and Functions Test

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Relations and Functions Test - 56

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Weekly Quiz Competition

Question 1
1 / -0

Find $$g(x)$$, if $$f(x) = 7x + 12$$ and $$f(g(x) = 21x^{2} + 40$$

A
$$21x^{2} + 28$$

B
$$21x^{2}$$

C
$$7x^{2} + 4$$

D
$$3x^{2} + 28$$

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

Solution

Given, $$f(x) = 7x+12, f(g(x))=21x^2+40$$
$$\therefore f(g(x)) = 7g(x)+12 = 21 {x}^{2} + 40$$
$$\therefore 7 g(x) = 21 {x}^{2}+28$$
$$\therefore g(x) = 3 {x}^{2} + 4$$
Question 2
1 / -0

Given a function $$f(x) = \log_{7}\dfrac {x}{8}$$ for $$x\geq 8$$, find $$f^{-1}(x) $$, for $$x$$ belonging to its domain.

A
$$8(7^{x})$$

B
$$7(8^{3})$$

C
$$8(7^{x +1})$$

D
$$\log_{4}\dfrac {x}{8}$$

E
$$\dfrac {x}{8}$$

Solution

Given $$f(x) = \log _{ 7 }{ \dfrac { x }{ 8 } } $$ , let it be $$y$$
We get $$f(x) = \log _{ 7 }{ \dfrac { x }{ 8 } } =y$$
$$\therefore x = f^{ -1 }\left( y \right) $$
Which implies $$ \log _{ 7 }{ \dfrac { x }{ 8 } } =y$$
$$\therefore \dfrac {x}{8} = {7}^{y}$$
We get $$x = 8({7}^{y}) = f^{ -1 }\left( y \right) $$
Now replace $$y$$ with $$x$$
Therefore $$f^{ -1 }\left( x \right) = 8({7}^{x})$$
Question 3
1 / -0

Given a function $$f(x) = \dfrac {1}{2}x - 4$$ and the composite function $$f(g(x)) = g(f(x))$$, determine which among the following can be $$g(x)$$:
I. $$2x - \dfrac {1}{4}$$
II. $$2x + 8$$
III. $$\dfrac {1}{2}x - 4$$

A
I only

B
II only

C
III only

D
II and III only

E
I, II, and III

Solution

If $$g(x)=2x-\dfrac14$$

$$g(f(x))=x-8-\dfrac14$$

$$f(g(x))=x-\dfrac18-4$$

Case II

$$g(f(x))=2(\dfrac{x}{2}-4)+8=x$$

$$f(g(x))=\dfrac12(2x+8)-4=x+4-4=x$$

Case III

$$g(f(x))=\dfrac12(\dfrac{x-8}{2})-4=\dfrac x4-6$$

$$f(g(x))=\dfrac12(\dfrac{x-8}{2})-4=\dfrac x4-6$$
Option D is correct
Question 4
1 / -0

If $$f(x) = 4x - 3$$ and $$g(x) = x - 4$$, determine which of the following composite function has a value of $$-11$$.

A
$$f(g(2))$$

B
$$g(f(2))$$

C
$$g(f(3))$$

D
$$f(g(3))$$

E
$$f(g(4))$$

Solution

Given, $$f(x)= 4x-3, g(x)=x-4$$
$$\Rightarrow f(g(2)) = f(-2) = -11$$
$$\Rightarrow g(f(2))=g(5) = 1$$
$$\Rightarrow g(f(3)) = g(9) = 5$$
$$\Rightarrow f(g(3)) = f(-1) = -7$$
$$\Rightarrow f(g(4)) = f(0) = -3$$
Question 5
1 / -0

The above figure shows the graph of the function $$f(x)$$, the value of $$f(f(3))$$ is:

A
$$-4$$

B
$$-2$$

C
$$0$$

D
$$1$$

E
$$3$$

Solution

From the graph shown, we need to find $$f(f(3))$$
The value of the function at $$x = 3$$ is given by $$f(3) = -2$$
Next, $$f(f(3)) = f(-2) = 0$$
Question 6
1 / -0

$$*$$ is a binary operation on $$Z$$ such that:
$$a * b = a + b + ab$$.
The solution of $$(3* 4) *x = -1$$ is

A
$$1$$

B
$$-1$$

C
$$4$$

D
$$3$$

Solution

$$(3*4)=(3+4)+(3\times 4)=7+12=19$$...(i)
Now
$$(3*4)*x=19*x=19+x+19x=-1$$ or $$20x=-20$$ or $$x=-1$$.
Hence the value of x is -1.
Question 7
1 / -0

Extraction of a cube root of a given number is

A
binary operation

B
relation

C
unary operation

D
relation in some set

Solution

Extraction of cube root of a given number is binary operation
Therefore the correct option is $$A$$
Question 8
1 / -0

If $$f(x) = 3x - 5$$ and $$g(x) = x^2 + 1, f [g(x)] =$$

A
$$3x^2-5$$

B
$$3x^2+6$$

C
$$x^2-5$$

D
$$3x^2-2$$

E
$$3x^2+5x-2$$

Solution

Given $$f(x) = 3x-5$$ and $$g(x) = {x}^{2}+1$$
$$\therefore f(g(x)) = f({x}^{2}+1) $$
$$= 3({x}^{2}+1)-5 $$
$$= 3{x}^{2}-2$$
Question 9
1 / -0

If $$f(x)=3x-1$$ and if $${f}^{-1}$$ is the inverse function of $$f$$, what is $${f}^{-1}(5)$$?

A
$$18$$

B
$$\cfrac{4}{3}$$

C
$$\cfrac{5}{3}$$

D
$$2$$

E
$$0$$

Solution

$$y=3x-1$$

$$\Rightarrow x=\dfrac{y+1}3$$

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

$$f^{-1}(5)=\dfrac{5+1}3=2$$
Question 10
1 / -0

If $$a*b = |a-b|$$ then $$6*8$$ will be

A
$$-2$$

B
$$14$$

C
$$2$$

D
Cannot be determined

Solution

$$\mathbf{{\text{Step -1: Identifying the binary function definition}}{\text{.}}}$$

$${\text{In this * is a binary function, which performs an operation defined as}}$$

  $${\text{If the operation is performed between two variables a and b such as }a*b \text{ then}}$$

  $${{a*b = |a - b| }}{\text{ i.e}}{\text{. the binary operation between two variables will return the }}$$

  $${\text{absolute difference between them}}.$$

$$\mathbf{{\text{Step -2: Finding the value of 6*8}}{\text{.}}}$$

  $${\text{Working according to the definiton of the operation we can write,}}$$

  $${{6*8 = |6 - 8| = | - 2| = 2}}$$

$$\mathbf{{\text{Therefore, the value of 6*8 = 2}}{\text{.}}}$$

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Relations and Functions Test - 56

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Relations and Functions Test - 56

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Question 1

$$21x^{2} + 28$$

$$21x^{2}$$

$$7x^{2} + 4$$

$$3x^{2} + 28$$

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

Solution

Question 2

$$8(7^{x})$$

$$7(8^{3})$$

$$8(7^{x +1})$$

$$\log_{4}\dfrac {x}{8}$$

$$\dfrac {x}{8}$$

Solution

Question 3

I only

II only

III only

II and III only

I, II, and III

Solution

Question 4

$$f(g(2))$$

$$g(f(2))$$

$$g(f(3))$$

$$f(g(3))$$

$$f(g(4))$$

Solution

Question 5

$$-4$$

$$-2$$

$$0$$

$$1$$

$$3$$

Solution

Question 6

$$1$$

$$-1$$

$$4$$

$$3$$

Solution

Question 7

binary operation

relation

unary operation

relation in some set

Solution

Question 8

$$3x^2-5$$

$$3x^2+6$$

$$x^2-5$$

$$3x^2-2$$

$$3x^2+5x-2$$

Solution

Question 9

$$18$$

$$\cfrac{4}{3}$$

$$\cfrac{5}{3}$$

$$2$$

$$0$$

Solution

Question 10

$$-2$$

$$14$$

$$2$$

Cannot be determined

Solution

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