Relations and Functions Test

Result

Relations and Functions Test - 79

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Question 1
1 / -0

If $$f(x)=x-\cfrac{1}{x}$$ then number of solutions of $$f(f(f(x)))=1$$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$
Question 2
1 / -0

Show that the function $$f:[0, \infty)\rightarrow [0, \infty)$$ defined by $$f(x)=\dfrac{2x}{1+2x}$$ is?

A
One-one and onto

B
One-one but not onto

C
Not one-one but onto

D
Neither one-one nor onto

Solution
Question 3
1 / -0

If $$f(x)=1+|x-1|,-1 \le x \le 3$$ and $$g(x)=2-|x+1|,-2 \le x \le 2$$ then choose the appropriate option.

A
$$fog(x)=x-1$$ for $$x\ \in\ (0,1)$$

B
$$fog(x)=x$$ for $$x\ \in\ (-1,1)$$

C
$$gog(x)=x$$ for $$x\ \in\ (-1,2)$$

D
$$all\ of\ these$$

Solution
Question 4
1 / -0

Let $$f(x)=\dfrac{x^{2}-4}{x^{2}+4}$$ for $$|x|>2$$, then the function $$f:(-\infty, -2)\cup [2,\infty)\rightarrow (-1,1)$$ is

A
One-one into

B
One-one onto

C
Many one into

D
Many one onto

Solution

$$\begin{array}{l} f\left( x \right) =\frac { { { x^{ 2 } }-4 } }{ { { x^{ 2 } }+4 } } \\ let\, \, f\left( x \right) =y \\ \Rightarrow y=\frac { { { x^{ 2 } }-4 } }{ { { x^{ 2 } }+4 } } \\ \Rightarrow { x^{ 2 } }y+4y-{ x^{ 2 } }+4=0 \\ \Rightarrow { x^{ 2 } }\left( { y-1 } \right) +4\left( { y+1 } \right) =0 \\ \Rightarrow { x^{ 2 } }=\frac { { -4\left( { y+1 } \right) } }{ { y-1 } } \\ \Rightarrow x=\frac { { \sqrt { 4\left( { y+1 } \right) } } }{ { \left( { 1-y } \right) } } \\ \Rightarrow \frac { { 4\left( { y+1 } \right) } }{ { 1-y } } 0 \\ \Rightarrow y+,0 \\ \Rightarrow y-1 \\ \Rightarrow 1-y0 \\ \Rightarrow y1 \\ Hence,\, \, range\, \, of\, \, { f^{ b } }\, \, is\left( { -1,1\, \, which\, \, is\, \, equal\, \, to\, \, Codomain } \right) \\ \therefore \, \, { f^{ h } }\, \, is\, \, one-one-onto{ 1mu } \end{array}$$
Question 5
1 / -0

Let $$f:R\rightarrow R$$ be defined by $$f(x)=\dfrac {x|x|}{2}+\cos x+1$$ then $$f(x)$$ is

A
One-one only

B
Onto only

C
Neither one-one nor onto

D
Bijection

Solution
Question 6
1 / -0

Let $$f : R \rightarrow (-1,1)$$ be defined as $$f(x)=\dfrac {e^{x}-e^{-x}}{e^{x}+e^{-x}}$$ then $$f$$ is

A
One-one onto

B
One-one into

C
Many-one onto

D
Many-one-into

Solution
Question 7
1 / -0

$$f:A \rightarrow A,A=\left\{a_{1},a_{2},a_{3},a_{4},a_{5}\right\}$$, then the number of one one function so that $$f(x_{i})\neq x_{i},x_{i}\ \in\ A$$ is

A
$$44$$

B
$$88$$

C
$$22$$

D
$$20$$

Solution
Question 8
1 / -0

If $$P(S)$$ denotes the set of all subsets of a given set S, then the number of one-to-one functions from the set $$S=\{1,2,3\}$$ to he set $$P(S)$$

A
$$24$$

B
$$8$$

C
$$336$$

D
$$320$$

Solution
Question 9
1 / -0

$$\begin{aligned} \text { If } A & = \{ x | x / 2 \in Z , 0 \leq x \leq 10 \} \\ B & = \{ x | x \text { is one digit prime } \} \\ C & = \{ x | x / 3 \in N , x \leq 12 \} \end{aligned}$$,
Then $$A \cap ( B \cup C )$$ is equal to-

A
$$\{ 2,6 \}$$

B
$$\{3,6,12 \}$$

C
$$\{ 2,6,12 \}$$

D
$$\{ 6,8 \}$$

Solution
Question 10
1 / -0

The function $$f:N\rightarrow N $$ defined by $$f\left( x \right) =x-5\left[ \dfrac { x }{ 5 } \right]$$, where $$N$$ is the set of natural numbers and $$[x]$$ denotes the greatest integer less then or equal to $$x$$ is

A
One-one and onto

B
One-one but not onto

C
Onto but not one-one

D
Neigher one-one nor onto

Solution

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