Let $f(x)$ be a polynomial with integer coefficients satisfying $f(1)=5$ and $f(2)=7$. The smallest possible positive value of $f(12)$ 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?

The relation $f$ is defined by $f(x) = \begin{cases} x^2, & 0 \le x \le 3 \\ 3x, & 3 \le x \le 10 \end{cases}$. The relation $g$ is defined by $g(x) = \begin{cases} x^2, & 0 \le x \le 2 \\ 3x, & 2 \le x \le 10 \end{cases}$. Show that $f$ is a function and $g$ is not a function.

$A$ function $f$ is defined by $f(x) = 2x - 5$. Find the values of $f(0)$,$f(7)$,and $f(-3)$.

Let $f: X \rightarrow Y$ be a function and $A_y = f^{-1}(\{y\})$ for $y \in Y$. Then $A_i \cap A_j = \phi$ $(i \neq j)$ for all $i, j \in Y$ and $\bigcup_{y \in Y} A_y = X$,if

Let $N$ be the set of natural numbers. Define a function $f: N \rightarrow N$ by $f(x) = 2x + 1$. Using this definition,complete the table given below.

$x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$y$	$f(1) = \dots$	$f(2) = \dots$	$f(3) = \dots$	$f(4) = \dots$	$f(5) = \dots$	$f(6) = \dots$	$f(7) = \dots$