Real Valued Functions

Q. Consider functions $f:A\to B$ and $g:B\to C$ $(A,B,C\subseteq R)$ such that $(gof{)}^{-1}$ exists, then

1. $f$ is one-one and $g$ is onto
2. $f$ is onto and $g$ is one-one
3. $f$ and $g$ both are one-one
4. $f$ and $g$ both are onto

Q. If the domain of the function $f$ is $R$, then which of the following will not be a real valued function ?

1. $f\left(x\right)=|x-2|$
2. $f\left(x\right)=x^2$
3. $f\left(x\right)=5x-7$
4. $f\left(x\right)=\sqrt{x}$

Q. The domain of the function $f\left(x\right)=\sqrt{10-\sqrt{x^4-21x^2}}$ is

1. $[5,\infty )$
2. $[-5,-\sqrt{21}]\cup [\sqrt{21},5]\cup \left\{0\right\}$
3. $[-\sqrt{21},\sqrt{21}]$
4. $(-\infty ,-5)$

Q. Let $f:N\to R$ be a function satisfying $f\left(1\right)+2f\left(2\right)+3f\left(3\right)+4f\left(4\right)+\dots +nf\left(n\right)=n(n+1)f\left(n\right)$ for $n\ge 2$. If $f\left(1\right)=1$, then which of the following is/are true?

1. $f\left(1003\right)=\frac{1}{2006}$
2. $f\left(999\right)=\frac{1}{1998}$
3. $f\left(1998\right)=\frac{1}{1998}$
4. $f\left(2\right),f\left(3\right)$ and $f\left(4\right)$ are in H.P.

Q.

If $f\left(x\right)=\sin \sqrt{x}$, then period of $f\left(x\right)$ is

1. $\mathrm{\pi }$

2. $\frac{\mathrm{\pi }}{2}$

3. $2\mathrm{\pi }$

4. None of these.

Q. The number of solution(s) of the equation $\text{sgn}\left(\ln x\right)=3$ is
(Here, sgn denotes the signum function)

1. $4$
2. $0$
3. $3$
4. $2$

Q.

The domain of the function $f\left(x\right)=\sqrt{2-2x-x^2}$ is

1. $[-\sqrt{3},\sqrt{3}]$

2. $[-2,2]$

3. $[-2-\sqrt{3},-2+\sqrt{3}]$

4. $[-1-\sqrt{3},-1+\sqrt{3}]$

Q. Let $S=\{\theta \in [-2\pi ,2\pi ]:2{\cos }^2\theta +3\sin \theta =0\}$.
Then the sum of the elements of $S$ is:

1. $\frac{13\pi }{6}$
2. $\frac{5\pi }{3}$
3. 2$\pi$
4. $\pi$

Q. Which of the following is (are) CORRECT for a real function?

1. Domain of a real function should be subset of $R$
2. Range of a real function should be subset of $R$
3. If $f:R\to R$ and $f\left(x\right)=1-x^2$, then it is real function
4. If $f:R\to R$ and $f\left(x\right)=x^3-2|x|$, then it is real function

Q.

The range of$f\left(x\right)=\cos x-\sin x$ is

1. $[-1,1]$

2. $(-1,2)$

3. $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$

4. $\left(-\sqrt{2},\sqrt{2}\right)$

Q.

Find gof and fog, if
(i) f(x) =|x| and g(x) =|5x-2|

$\left(ii\right)f\left(x\right)=8x^3$ and $g\left(x\right)=x^{\frac{1}{3}}$.

Q. Number of solution(s) of the equation $x^2-5x\text{sgn}(x^2-9)+6=0$ is
(Here, $\text{sgn}$ denotes the signum function)

1. $0$
2. $2$
3. $3$
4. $1$

Q.

The function $f\left(x\right)={\cot }^{-1}\left(x\right)+x$ increases in the interval

1. $(1,\infty )$

2. $(-1,\infty )$

3. $(-\infty ,\infty )$

4. $(0,\infty )$

Q. Let $f\left(x\right)$ be a real valued function. Then the domain of $f\left(x\right)=\sqrt{x-\sqrt{1-x^2}}$ is

1. $[-1,-\frac{1}{\sqrt{2}}]\cup [\frac{1}{\sqrt{2}},1]$
2. $(-\infty ,-\frac{1}{\sqrt{2}}]\cup [\frac{1}{\sqrt{2}},\infty )$
3. $[-1,1]$
4. $[\frac{1}{\sqrt{2}},1]$

Q. If $x$ is rational, $f\left(x\right)=1$ else $f\left(x\right)=0$. Then $fof\left(\sqrt{5}\right)=$

Q. If A = $\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$, find $\alpha$ satisfying $0<\alpha <\frac{\pi }{2}$ when $A+A^T=\sqrt{2}{I}_2$; where $A^T$ is transpose of A.

Q.

Let $f\left(x\right)=\sqrt{x^2+1}$. Then, which of the following is correct?

1. $f\left(xy\right)=f\left(x\right)f\left(y\right)$

2. $f\left(xy\right)\ge f\left(x\right)f\left(y\right)$

3. $f\left(xy\right)\le f\left(x\right)f\left(y\right)$

4. None of these

Q.

Let $\left[.\right]$ denote the greatest integer function $f\left(x\right)=\left[{\tan }^{2}x\right]$, then:

1. $li{m}_{x\to 0}f\left(x\right)$ does not exist

2. $f\left(x\right)$ is continuous at $x=0$

3. $f\left(x\right)$ is not differentiable at $x=0$

4. $f^{}\left(x\right)=1$

Q.

Let $b$ be a nonzero real number. Suppose $f=R\to R$ is a differentiable function such that $f\left(0\right)=1$. If the derivative $f^{}$of $f$ satisfies the equation $f^{}\left(x\right)=\frac{f\left(x\right)}{\left(b^2+x^2\right)}$. For all $x\in R$, then which of the following statements is/are true $?$

1. If $b>0$, then $f$ is an increasing function

2. If $b<0$, then $f$ is a decreasing function

3. $f\left(x\right)f\left(-x\right)=1$ for all $x\in R$

4. $f\left(x\right)-f\left(-x\right)=0$ for all $x\in R$

Q. The domain of the function $f\left(X\right)=\sqrt{3-2^x-2^{1-x}}+\sqrt{si{n}^{-1}x}$ is

1. [0, 1]
2. [-1, 0]
3. [1, 2]
4. $[\frac{1}{2},1]$

Q. Range of function $y={\log }_2\left(\sin x\right)$ is :

1. $(-\infty ,0)$
2. $(0,\infty )$
3. $(-\infty ,0]$
4. $R$

Q. The number of real solution(s) of the equation $\text{sgn}(2x-4)=2$ is
(Here, sgn denotes the signum function)

Q. If $x^2-3x\text{sgn}\left(x\right)+2=0,$ then the value(s) of $x$ is/are
(where $\text{sgn}\left(x\right)$ denotes the signum function)

1. $-1$
2. $-2$
3. $1$
4. $2$

Q.

If $\mathrm{z}\left(1+\mathrm{a}\right)=\mathrm{b}+\mathrm{ic}$ and ${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=1,$ then $\frac{\left(1+\mathrm{iz}\right)}{\left(1-\mathrm{iz}\right)}$ is equal to

1. $\frac{\left(\mathrm{a}+\mathrm{ib}\right)}{\left(1+\mathrm{c}\right)}$

2. $\frac{\left(\mathrm{b}-\mathrm{ic}\right)}{\left(1+\mathrm{a}\right)}$

3. $\frac{\left(\mathrm{a}+\mathrm{ic}\right)}{\left(1+\mathrm{b}\right)}$

4. None of these

Q. If $f\left(x\right)=x^2-5x+6$ is a real-valued function, then

1. $f\left(x\right)\ge 0$ when $x\in (-\infty ,2]\cup [3,\infty )$
2. $f\left(x\right)\ge 0$ when $x\in [2,\infty )$
3. $f\left(x\right)<0$ when $x\in (-\infty ,2)$
4. $f\left(x\right)<0$ when $x\in (2,3)$

Q. What is the approximate value of $\sqrt[5]{5242.999}?$

1. $\frac{1214999}{4050}$
2. $\frac{1115}{405}$
3. $\frac{1214999}{405000}$
4. $\frac{1214999}{40500}$

Q.

If f is a real function satisfying $f(x+\frac{1}{x})=x^2+\frac{1}{x^2}$ for all $xϵR-\left\{0\right\}$, then write the expression for f(x).

Q.

Is a hyperbola an odd function?

Q. Number of solution(s) of the equation $x^2-5x\text{sgn}(x^2-9)+6=0$ is
(Here, $\text{sgn}$ denotes the signum function)

1. $0$
2. $2$
3. $3$
4. $1$

Q. Which of the following functions are the mirror images of each other with respect to the line $y=x$

1. $e^x$ and ${\log }_{e}x$
2. $e^x$ and ${\log }_{x}e$
3. ${\pi }^x$ and ${\log }_{\pi }x$
4. ${\pi }^x$ and ${\log }_x\pi$