Real Valued Functions

Q. If the graph of function $f\left(x\right)=a^x$ is


Then $a$ will be

1. $a>0$
2. $a>1$
3. $0<a<1$
4. $a<0$

Q. If the function $f\left(x\right)=\sqrt{{0.25}^{4-3x}-4^{2x}}$ is defined, then the possible of $x$ is/are

1. $2.5$
2. $3.8$
3. $4.7$
4. $6$

Q. If $x-0.5=\text{sgn}\left(x\right),$ then the possible value(s) of $x$ is/are
(where $\text{sgn}\left(x\right)$ denotes the signum function)

1. $0.5$
2. $-0.5$
3. $1.5$
4. $-1.5$

Q. Number of values of $x$, satisfying the equation $\sqrt{(x+8)+2\sqrt{(x+7)}}+\sqrt{(x+1)-\sqrt{x+7}}=4,$ is

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

Q. Which of the following option(s) is/are correct for exponential function $f\left(x\right)=a^x$ ?

1. Domain of $f\left(x\right)$ is $R$
2. Range of $f\left(x\right)$ is $(0,\infty )$
3. $a\in R-\{0,1\}$
4. $a\in (0,1)\cup (1,\infty )$

Q.

If $f\left(x\right)=\frac{x^2-x}{x^2+2x}$, then the domain of $f$ is

1. $\left\{0,-2\right\}$

2. $R-\left\{0,-2\right\}$

3. $R$

4. None of these

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 ${\log }_x\pi$

Q. If $x-0.5=\text{sgn}\left(x\right),$ then the possible value(s) of $x$ is/are
(where $\text{sgn}\left(x\right)$ denotes the signum function)

1. $0.5$
2. $-0.5$
3. $1.5$
4. $-1.5$

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

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

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

Q. Let $f\left(x\right)$ be a real valued function, then the domain of $f\left(x\right)=\sqrt{x-4-2\sqrt{x-5}}-\sqrt{x-4+2\sqrt{x-5}}$ is

1. $(5,\infty )$
2. $[5,\infty )$
3. $(4,\infty )$
4. $[4,\infty )$

Q. Which of the following options is/are true for the function $f\left(x\right)=|x-3|+|x-4|$

1. The minimum value of $f\left(x\right)$ is $1$
2. $f\left(x\right)=2x-7$ when $x\ge 4$
3. $f\left(x\right)=1$ when $x<3$
4. The graph of $f\left(x\right)$ is
   

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 number of values of $x$ satisfying the equation $1+x=\text{sgn}\left(x\right)$ is
(where $\text{sgn}\left(x\right)$ denotes the signum function)

Q. $\mathrm{If}f\left(x\right)=\cos \left(\log x\right),x>0;\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}f\left(x\right)f\left(4\right)-\frac{1}{2}\left(f\left(\frac{x}{4}\right)+f\left(4x\right)\right\}=$

1. 1
2. -1
3. \N
4. $\pm 1$

Q. If the graph of function $f\left(x\right)=a^x$ is


Then $a$ will be

1. $a>0$
2. $a>1$
3. $0<a<1$
4. $a<0$

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. $2$