Real Valued Functions

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. Consider the real-valued function $f$ satisfying $2f\left(\sin x\right)+f\left(\cos x\right)=x$. Then,

1. Domain of $f$ is $R$
2. Domain of $f$ is $[-1,1]$
3. Range of $f$ is $[-\frac{2\pi }{3},\frac{\pi }{3}]$
4. Range of $f$ is $R$

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 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. Which of the following is/are a function?

1. $\left\{\right(x,y)|x,y\in R,x^2=y\}$
2. $\left\{\right(x,y)|x,y\in R,y^2=x\}$
3. $\left\{\right(x,y)|x,y\in R,x=y^3\}$
4. $\left\{\right(x,y)|x,y\in R,y=x^3\}$

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. 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. 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 $x^2-3x\text{sgn}\left(x\right)+2=0,$ 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. 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. $a\in (0,1)\cup (1,\infty )$