Functions Test 51

Result

Functions Test 51

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Question 1
1 / -0

A real valued function $$f(x)$$ satisfies the function equation $$f(x-y)=f(x)f(y)-f(a-x)f(a+y)$$ where a is a given constant and $$f(0)=1, f(2a-x)$$ is equal to?

A
$$f(A)+f(a-x)$$

B
$$f(-x)$$

C
$$-f(x)$$

D
$$f(x)$$

Solution
Question 2
1 / -0

Let $$f\left( x \right)={2^{10}}x + 1$$ and $$g\left( x \right) = {3^{10}}x - 1$$ , If $$\left( {fog} \right)\left( x \right) = x$$ , then $$x$$ is equal to

A
$$\dfrac{{{3^{10}} - 1}}{{{3^{10}} - {2^{ - 10}}}}$$

B
$$\dfrac{{{2^{10}} - 1}}{{{2^{10}} - {3^{ - 10}}}}$$

C
$$\dfrac{{1 - {3^{10}}}}{{{2^{10}} - {3^{ - 10}}}}$$

D
$$\dfrac{{1 - {2^{-10}}}}{{{3^{10}} - {2^{ - 10}}}}$$

Solution

Given:$$f\left(x\right)={2}^{10}x+1$$ and $$g\left(x\right)={3}^{10}x-1$$

$$f\left(g\left(x\right)\right)=x$$

$$\Rightarrow\,f\left({3}^{10}x-1\right)=x$$

$$\Rightarrow\,{2}^{10}\left({3}^{10}x-1\right)+1=x$$

$$\Rightarrow\,{2}^{10}{3}^{10}x-{2}^{10}+1=x$$

$$\Rightarrow\,-{2}^{10}+1=x-{2}^{10}{3}^{10}x$$

$$\Rightarrow\,-{2}^{10}+1=x\left(1-{2}^{10}{3}^{10}\right)$$

$$\Rightarrow\,x=\dfrac{-{2}^{10}+1}{1-{2}^{10}{3}^{10}}$$

$$\Rightarrow\,x=\dfrac{{2}^{10}-1}{{2}^{10}{3}^{10}-1}$$

$$\Rightarrow\,x=\dfrac{{2}^{10}-1}{{2}^{10}{3}^{10}-1}$$

$$\Rightarrow\,x=\dfrac{{2}^{10}-1}{{2}^{10}\left({3}^{10}-{2}^{-10}\right)}$$

$$\Rightarrow\,x=\dfrac{1-{2}^{-10}}{\left({3}^{10}-{2}^{-10}\right)}$$
Question 3
1 / -0

$$f : R^+ \rightarrow R$$ defined by $$f(x) = 2^x , \, x \in (0, 1), \, f(x) = 3^x , \, x \in [1, \, \infty)$$ is

A
onto

B
one-one

C
neither one-one nor onto

D
one one onto

Solution
Question 4
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 5
1 / -0

If $$f(x)=|x|$$ and $$g(x)=[x]$$, then value of $$fog \left(-\dfrac {1}{4}\right)+gof \left(-\dfrac {1}{4}\right)$$ is ?

A
$$0$$

B
$$1$$

C
$$-1$$

D
$$1/4$$

Solution
Question 6
1 / -0

The function $$f : R\rightarrow R$$ given by, then $$f(x)=3-2\sin x$$ is

A
one-one

B
onto

C
bijective

D
None of these

Solution
Question 7
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 8
1 / -0

If $$f\left( x \right) = \sin ^{ 2 }{ x } + \sin ^{ 2 }({ { x }+\frac { \pi }{ 3 }) + \cos { x\cos { \left( { x } + \frac { \pi }{ 3 } \right), ~g(\frac { 5 }{ 4 }) = 1, \text{then} \left( gof \right)\left( x \right) } }\ \text{ is}\ \text{equal}\ \text{to} } $$

A
$$1$$

B
$$0$$

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

D
$$\frac{1}{2}$$

Solution
Question 9
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 10
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
$$4$$