Definition of Function

Q. The domain of the function
$f\left(x\right)={\sin }^{-1}\left(\frac{x^2+1}{2x}\right)+\frac{1}{\text{log}\{2-x\}}$
(where $(.)$ denotes the fractional part function) is

1. $\Phi$ (empty set)
2. $\{x:x\in I\text{and}x\le 4\}$
3. $(-\infty ,0)\cup (0,1)$
4. $\{x:x\in I\text{and}x\ge 3\}$

Q. Which of the following is not a function?

1. 
2. 
3. 
4. 

Q. Which of the following is/are a function?

1. 
2. 
3. 
4. 

Q. Which of the following is/are functions?

1. 
2. 
3. 
4. 

Q. A relation defined from $A=\{0,1,2,3,4\}$ to $N$ as $xRy\text{iff}x=y$. Then which among the following options is correct

1. $R^{-1}$ is a function with range as $A$
2. $R$ is a function with range as $N$
3. $R$ is a function with range as $\{1,2,3,4\}$
4. $R$ is not a function

Q. If $n\left(A\right)=n,n>0$ then which among the following statements are correct

1. Number of relations on $A$ that are not reflexive $=\left(2^{n^2-n}\right)(2^n-1)$
2. Number of relations on $A$ that are not reflexive $=\left(2^{n^2+n}\right)(2^n-1)$
3. Number of relations on $A$ that are not symmetric
   $=2^{n^2}-2^{\frac{n(n+1)}{2}}$
4. Number of relations on $A$ that are not symmetric
   $=2^{n^2}-2^{\frac{n(n-1)}{2}}$

Q. Which among the following relations between $x,y$ doesn't represent a function

1. $(x-1{)}^2+(y-3{)}^2=5^2,x\in [-4,6],y\in R$
2. $y^2=4x,x\in [0,\infty ),y\in R$
3. $x^2=4y,x,y\in R$
4. $y=2x+5^2,x,y\in R$

Q. Let $a,b,c$ be rational numbers and $f:Z\to Z$ be a function given by
$f\left(x\right)=a{x}^2+bx+c$ then $a+b$ is

1. An integer
2. A rational number but not an integer
3. Negative integer
4. Nothing in particular can be said

Q. If $A=\{1,2,3,4\}$ then which among the following relations are not functions from $A$ to itself

1. $f_1=\left\{\right(x,y)|y=x+1,\forall x,y\in A\}$
2. $f_2=\left\{\right(x,y)|x+y>4,\forall x,y\in A\}$
3. $f_3=\left\{\right(x,y)|y<x,\forall x,y\in A\}$
4. $f_4=\left\{\right(x,y)|x+y=5,\forall x,y\in A\}$
   

Q. Which among the following is\are function(s)

1. 
2. 
3. 
4.