$A$ function $f$ is defined by $f(x) = 2x - 5$. Find the value of $f(7)$.

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:

$A$ function $f$ is defined by $f(x) = 2x - 5$. Find the value of $f(-3)$.

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.

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

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 ${}^qC_p$.