Domain

Q. The domain of the function $f\left(x\right)=x^{1/\ln x}$ is

1. $(0,\infty )-\{1,e\}$
2. $(e,\infty )$
3. $(0,1)\cup (1,\infty )$
4. $(1,\infty )-\left\{e\right\}$

Q. Let $f\left(x\right)=\frac{x^2-1}{x},g\left(x\right)=\frac{x+2}{x-3}$ then domain of $\frac{f\left(x\right)}{g\left(x\right)}$ is

1. $R-\{0,-2\}$
2. $R-\{-2,0,3\}$
3. $R$
4. $R-\{0,-3\}$

Q. Consider the set $A=\{1,2,3\}$ and the relation on $A$ as $R=\left\{\right(1,2),(1,3\left)\right\},$ then $R$ is

1. a reflexive relation
2. a symmetric relation
3. None of the above
4. a transitive relation

Q. A relation $R$ is defined on the set of integers as follows : $(a,b)\in R\iff a^2+b^2=25.$
Then domain of $R$ is

1. $\{3,4,5\}$
2. $\{0,3,4,5\}$
3. $\{0,\pm 3,\pm 4,\pm 5\}$
4. $\{\pm 3,\pm 4,\pm 5\}$

Q. Let $f:D\to \{1,3,9,19\}$ be a function defined as $f\left(x\right)=2x^2+1$, where $D$ is the domain of $f$. If range and co-domain of the function $f$ are equal, then the maximum number of elements in set $D$ can be

Q. Let $A=\{x:x\in Z,|x|\le 3\}$ and $B=\{y;y\in N,-1<y+2\le 4\}$. If $R$ be the relation from $A$ to $B$, then

1. Co domain of $R$ is $\{1,2\}$
2. Range of $R$ is $\{0,1,2\}$
3. $R^{-1}$ can be $\left\{\right(1,1),(1,2),(1,3),(2,1),(2,2),(2,3\left)\right\}$
4. Co domain of $R^{-1}$ is $\{-3,-2,-1.0,1,2,3\}$

Q. The characteristic of logarithm to the base $10$ of $0.000234$ is

Q. If $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$, where $f\left(x\right)=x^2-2x$ and $g\left(x\right)=x^3-3x^2+2x,$ then the domain of $h\left(x\right)$ is

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

Q. Let $X=\{p,q,r\},$ $Y=\{a,b,c\}$ and $Z=\{5,6,7\}$. Consider the relation $R_1$ from $X$ to $Y$ and $R_2$ from $Y$ to $Z$ such that $R_1=\left\{\right(p,a),(q,b),(r,c\left)\right\}$
$R_2=\left\{\right(a,5),(b,6),(c,7\left)\right\}$
Then which of the following is/are TRUE?

1. Range of $R_{2}o{R}_1$ is $\{5,6,7\}$
2. Domain of $R_{2}o{R}_1$ is $\{p,q,r\}$
3. Domain of $R_{2}o{R}_1$ is $\{a,b,c\}$
4. Range of $R_{2}o{R}_1$ is $\{a,b,c\}$

Q. The domain of the function
$f\left(x\right)=x\cdot \frac{1+2(x+4{)}^{-0.5}}{2-(x+6{)}^{0.5}}+(x+5{)}^{0.5}+4(x+10{)}^{-0.5}$ is

1. $(-4,\infty )-\left\{2\right\}$
2. $(-4,\infty )-\{-2\}$
3. $R-\left(-4,0\right)$
4. $(0,\infty )$

Q. Let $f\left(x\right)=\frac{3}{x-2}-\frac{1}{x+3}$ and $g\left(x\right)=\frac{x^2-4x+19}{x^2+x-6}$. If $f\left(x\right)=g\left(x\right)$, then $x=$

1. $2$
2. $3$
3. $4$
4. $6$

Q. Let $f:(-\infty ,+1]\to R,g:[-1,\infty )\to R$ be such that $f\left(x\right)=\sqrt{1-x}$ and $g\left(x\right)=\sqrt{1+x}$, then $f\left(x\right)+\frac{1}{g\left(x\right)}$ exist if $x\in$

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

Q. The domain of the function $f\left(x\right)=\sqrt{(x^2-5x+6)}+\sqrt{(8-x^2+2x)}$ is

1. $[-2,4]$
2. $[-2,2]\cup [3,4]$
3. $[2,4]$
4. $(-\infty ,2)$

Q. The function : $R\to [-\frac{1}{2},\frac{1}{2}]$ defined as $f\left(x\right)=\frac{x}{1+x^2}$ is

1. surjective but not injective
2. neither injective nor surjective
3. invertible
4. injective but not surjective

Q. The range of the function $f\left(x\right)={\log }_2(3-2x-x^2)$ is

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

Q. If $f\left(x\right)=\frac{x^2-3x+1}{x^2+4x+3},$ then domain of $f\left(x\right)$ is

1. $R-\{-1,3\}$
2. $R$
3. $R-\{1,3\}$
4. $R-\{-1,-3\}$

Q. The domain of $f\left(x\right)=\sqrt{\frac{4-x^2}{\left[x\right]+2}}$ is
(where [.] represents the greatest integer function)

1. $(-\infty ,1)$
2. $(-\infty ,-2)\cup [-1,2]$
3. $(-\infty ,-1)\cup [2,\infty )$
4. $(-\infty ,1)\cup [2,\infty )$

Q. Let $f\left(x\right)=f_1\left(x\right)-2f_2\left(x\right)$,

where, $f_1\left(x\right)=\{\begin{array}{cc}min\{x^2,|x|\},& |x|\le 1\\ max\{x^2,|x|\},& |x|>1\end{array}$

and, $f_2\left(x\right)=\{\begin{array}{cc}min\{x^2,|x|\},& |x|>1\\ max\{x^2,|x|\},& |x|\le 1\end{array}$

and, $g\left(x\right)=\{\begin{array}{c}min\left\{f\right(t):-3\le t\le x,-3\le x<0\}\\ max\left\{f\right(t):0\le t\le x,0\le x\le 3\}\end{array}$

The graph of $y=g\left(x\right)$ in its domain is broken at

1. $1$ point
2. $2$ points
3. $3$ points
4. None of these

Q. If $x^2-6|x|+5\le 0$ and $|x+2|<3,$ then $x\in$

1. $[-5,-1]\cup [1,5]$
2. $[-5,-1]$
3. $(-5,-1]$
4. $(-5,-1]\cup [1,5)$

Q. The function $f\left(x\right)={\sin }^{-1}(x^2-2x+2)$ is defined at $x=a$ and $f\left(a\right)=b.$ Then

1. $b$ is an irrational number
2. $a$ is a negative integer
3. $b$ is a rational number
4. $a$ is a positive integer

Q. The domain of $\sqrt{x+2}+\frac{1}{{\log }_{10}(x+1)}$ is

1. $(0,\infty )$
2. $(-1,\infty )-\left\{0\right\}$
3. $(-1,0)\cup (1,\infty )$
4. $R-\{-2,-1\}$

Q. The domain of $f\left(x\right)={\left(\frac{2x+1}{x^2-10x-11}\right)}^{2020}$ is

1. $(0,\infty )$
2. $(-1,11)$
3. $R-\{-1,11\}$
4. $R-[-1,11]$

Q. Let ${}^{\prime }{f}^{\prime }$ be a real valued function such that $0\le f\left(x\right)\le \frac{1}{2}$ and for some fixed $a>0,f(x+a)=\frac{1}{2}-\sqrt{f\left(x\right)-\left(f\right(x){)}^2},\forall x\in R$, then the period of the function $f\left(x\right)$ is

1. $3a$
2. $a$
3. $2a$
4. $4a$

Q. If $f\left(x\right)$ is defined on $(-1,1)$ , then the domain of $g\left(x\right)=f\left(e^x\right)+f\left({\log }_e|x|\right))$ is

1. $(-e,-\frac{1}{e})$
2. $(0,\infty )$
3. $(e,\infty )$
4. $R-\left\{0\right\}$

Q.

If, $f\left(x\right)=cos\left[{\pi }^2\right]x+cos[-{\pi }^2]x,$ where [x] stands for greatest integer function, then

1. $f\left(\frac{\pi }{2}\right)=1$

2. $f\left(\pi \right)=1$

3. $f(-\pi )=0$

4. $f\left(\frac{\pi }{4}\right)=2$

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

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

Q. The domain of $f\left(x\right)=\sqrt{\frac{1-5^x}{7^{-x}-7}}$ is

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

Q. If $f\left(x\right)=\frac{7}{x}+\frac{2}{x+1}+\frac{1}{x-1},$ then domain of the function is

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

Q. The minimum value of $f\left(x\right)=(x-1{)}^2+(x-2{)}^2+\cdots +(x-10{)}^2$ occurs at $x=k.$ Then the value of $\left[k\right]$ is
$($where $[.]$ represents the greatest integer function$)$

Q.

Given that $g\left(x\right)=\left[f\right(x)-1{]}^2$. Find the domain of f(x) = 1 - 2x, given that $0\le g\left(x\right)<4$.

1. $[-1,3)$

2. $(-1,\frac{1}{2})$

3. $-1,1$

4. $(-3,1]$