The function $f(x) = x^{2} + bx + c$, where $b$ and $c$ are real constants, describes:

Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be the functions defined by $f(x) = \frac{x}{1+x^2}$ and $g(x) = \frac{x^2}{1+x^2}$. Then, the correct statement$(s)$ among the following is/are:

If $f: R \rightarrow R$,then the function $f(x) = x|x|$ is:

Let $A = \{x_1, x_2, \dots, x_7\}$ and $B = \{y_1, y_2, y_3\}$ be two sets containing seven and three distinct elements respectively. The total number of onto functions $f : A \to B$ such that there exist exactly three elements $x$ in $A$ with $f(x) = y_2$ is equal to:

The function $f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}} + \cos^3\left(\frac{x}{2}\right)$ from $R$ to itself is

Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of $2310$ and $f: A \rightarrow Z$ be the function $f(x) = \left[\log_2\left(x^2 + \left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is: