Relations and Functions Test

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Relations and Functions Test - 64

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Question 1
1 / -0

If $$g(f(x) ) = |\sin x |$$ and $$f(g(x))=(\sin\sqrt x)^2$$ , then

A
$$f(x) = \sin^2x. g(x) =\sqrt x$$

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

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

D
f and g can not be determined

Solution

$$g(f(x))=|sinx|=\sqrt{(sinx)^2}\\f(g(x))=(sin\sqrt x )^2=sin^2\sqrt x\\\therefore f(x)=sin^2x \>and \>g(x)=\sqrt x$$
Question 2
1 / -0

Let $$f$$, $$g:R\rightarrow {R}$$ be two functions defined as $$ f\left( x \right) =\left| x \right| +x$$, $$ g\left( x \right) =\left| x \right| -x, \forall x\in R$$. Then, find $$fog(x)$$

A
$$||x|-x|-|x|-x$$

B
$$||x|-x|+|x|-x$$

C
$$||x|-x|-|x|+x$$

D
None of thesse

Solution

$$f\left( x \right) = \left| x \right| + x$$
$$g\left( x \right) = \left| x \right| - x$$
$$fog\left( x \right) = f\left( {g\left( x \right)} \right)$$
$$ = f\left( {\left| x \right| - x} \right)$$
$$ = \left| {\left| x \right| - x} \right| + \left| x \right| - x$$
Question 3
1 / -0

Consider set $$A={1,2,3,4}$$ and set $$B={0,2,4,6,8}$$, then the number of one-one function from set $$A$$ to set $$B$$ is ?

A
$$5$$

B
$$24$$

C
$$120$$

D
None of these

Solution

Number of one-one function from
$$\underset{(m)}{A}$$ to $$\underset{(n)}{B} = \left\{\begin{matrix} \, ^nP_m, & if \, n \ge m \\ 0, & if \ n < m \end{matrix}\right.$$
$$m = 4, \ n = 5$$
One-one function $$=\, ^5P_4 = \dfrac{5!}{1!} = 120$$
Question 4
1 / -0

If the binary operation $$*$$ is on set of integers $$Z$$ is defined as
$$a * b = a + 2b ^{2}$$ , then the value of $$(8 * 3) * 2$$

A
$$26$$

B
$$22$$

C
$$32$$

D
$$34$$

Solution

$$a\ast b=a+2b^{2}$$
$$\Rightarrow (8\ast 3)\ast 2=(8+2(3)^{2})\ast 2$$
$$=(26\ast 2)$$
$$=26+2(2)^{2}$$
$$=26+8$$
$$=34$$
Question 5
1 / -0

If $$f(x)=2x+5$$ and $$g(x)=x^2+1$$ be two real function , then value of $$fog$$ at x=1

A
$$9$$

B
$$6$$

C
$$5$$

D
$$4$$

Solution
Question 6
1 / -0

Let $$f(x+\dfrac{1}{x})=x^2+\dfrac{1}{x^2}(x\neq 0)$$, then $$f(x)=$$

A
$$x^2$$

B
$$x^2-1$$

C
$$x^2-2$$

D
N.O.T

Solution

We are given
$$f(x+ \dfrac{1}{2})= x^{2}+ \dfrac{1}{x^{2}}$$ (where $$x \neq 0$$)

$$= x^{2}+ \dfrac{1}{x^{2}}+2-2$$.

$$=x^{2}+ \dfrac{1}{x^{2}}+ 2 (x^{2}) \left( \dfrac{1}{x^{2}} \right)-2$$.

$$f\left(x+ \dfrac{1}{x} \right)= \left(x+ \dfrac{1}{x} \right)^{2}-2$$

So, simply put $$x+ \dfrac{1}{x} \rightarrow x$$

we get $$f(x) = x^{2}-2$$
Question 7
1 / -0

If the function $$f(x)=ax+b$$ has its own inverse then the ordered pair (a, b) can be.

A
$$(1, 0)$$

B
$$(-1, 0)$$

C
$$(-1, 1)$$

D
$$(1, 1)$$

Solution

We are given that,
$$f(x)= f^{-1} (x) $$ or $$fof^{-1} (x)= x.$$
$$\therefore$$ now, For $$f^{-1} (x)$$
$$f(x)= y= ax+b.$$
$$\therefore x= \dfrac{y-b}{a}$$
$$\therefore f^{2} (x) = \dfrac{x-b}{a}$$
now, $$f(x) = f^{-1} (2) \Rightarrow ax+b= \dfrac{x-b}{a}$$
$$\therefore a^{2}x+ ab= x-b$$
$$\therefore a^{2}x- + ab+ b =0$$
$$\therefore (a^{2}-1) x+ b (x+1)= 0$$
$$x =\dfrac{-b {a+1}}{(a^{2}-1)}$$
$$\therefore = \dfrac{-b (a+1)}{(a+1)(a-1)} =\dfrac{-b}{a-1} = \dfrac{b}{1-a}$$
$$\therefore (1-a) x= b$$
For $$(A) a= 1 \Rightarrow b =0$$
$$(B) a=-1 \Rightarrow b= 2x$$
$$(C) a= -1 \Rightarrow b= 2x$$
$$(D) a= 1 \Rightarrow b =0$$.
So, the right answer is $$(a,b) = (1,0)$$
Question 8
1 / -0

Let $$E=\{1, 2, 3, 4\}$$ and $$F=\{1, 2\}$$ then the number of onto functions from E to F is

A
$$14$$

B
$$16$$

C
$$12$$

D
$$8$$

Solution

Number of onto function from $$E$$ to $$F.$$

$$E=\{1,2,3,4\}$$ $$F=\{1,2\} $$

Number of function from $$E$$ to $$F=2\times 2\times 2\times 2=16$$

We have to exclude functions where $$f(x)=1$$ & $$ f(x)=2$$.

$$\therefore$$ Total number of onto function $$=16-2=14$$.
Question 9
1 / -0

Let $$f\left( x \right) ={ x }^{ 2 },g\left( x \right) ={ 2 }^{ x }$$, then solution set of $$fog\left( x \right) =gof\left( x \right) $$ is

A
R

B
$$\left\{ 0 \right\} $$

C
$$\left\{ 0,2 \right\} $$

D
None of these

Solution
Question 10
1 / -0

Let $$f : R \rightarrow R$$ be defined by $$f(x) = x^2 - 3x + 4 $$ for all $$x \epsilon R$$, then $$f^{-1}(2)$$ is

A
2

B
1

C
Not defined

D
$$\dfrac{1}{2}$$

Solution

$$F:R\to R$$
$$f(x)=x^2-3x+4$$
$$f(1)=1-3+4$$
$$f(1)=2$$
$$\boxed {f^{-1}(2)=1}$$

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Relations and Functions Test - 64

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Relations and Functions Test - 64

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SHARING IS CARING

Question 1

$$f(x) = \sin^2x. g(x) =\sqrt x$$

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

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

f and g can not be determined

Solution

Question 2

$$||x|-x|-|x|-x$$

$$||x|-x|+|x|-x$$

$$||x|-x|-|x|+x$$

None of thesse

Solution

Question 3

$$5$$

$$24$$

$$120$$

None of these

Solution

Question 4

$$26$$

$$22$$

$$32$$

$$34$$

Solution

Question 5

$$9$$

$$6$$

$$5$$

$$4$$

Solution

Question 6

$$x^2$$

$$x^2-1$$

$$x^2-2$$

N.O.T

Solution

Question 7

$$(1, 0)$$

$$(-1, 0)$$

$$(-1, 1)$$

$$(1, 1)$$

Solution

Question 8

$$14$$

$$16$$

$$12$$

$$8$$

Solution

Question 9

R

$$\left\{ 0 \right\} $$

$$\left\{ 0,2 \right\} $$

None of these

Solution

Question 10

2

1

Not defined

$$\dfrac{1}{2}$$

Solution

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