Let $f: R \rightarrow R$ be defined by $f(x)=x^2+1$. Then,the pre-images of $17$ and $-3$ respectively are

Which of the following is $NOT$ a function?

Statement $1$ : If $A$ and $B$ are two sets having $p$ and $q$ elements respectively,where $q > p$. Then the total number of functions from set $A$ to set $B$ is $q^p$.
Statement $2$ : The total number of selections of $p$ different objects out of $q$ objects is ${}^qC_p$.

If $f(x) = x^4 - 2x^3 + 3x^2 - ax + b$ is divided by $(x - 1)$ and $(x + 1)$,the remainders are $5$ and $19$,respectively. If $f(x)$ is divided by $(x - 2)$,the remainder is:

If $Q$ denotes the set of all rational numbers and $f\left(\frac{p}{q}\right)=\sqrt{p^2-q^2}$ for any $\frac{p}{q} \in Q$, then observe the following statements.
$I$. $f\left(\frac{p}{q}\right)$ is real for each $\frac{p}{q} \in Q$.
$II$. $f\left(\frac{p}{q}\right)$ is a complex number for each $\frac{p}{q} \in Q$.
Which of the following is correct?

Let $A = \{1, 2, 3, 4\}$,$B = \{1, 5, 9, 11, 15, 16\}$ and $f = \{(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)\}$. Is $f$ a function from $A$ to $B$? Justify your answer.