The range of the function $f(x) = \left[ \frac{1}{\ln(x^2 + e)} \right] + \frac{1}{\sqrt{1 + x^2}}$ is,where $[*]$ denotes the greatest integer function and $e = \lim_{\alpha \to 0} (1 + \alpha)^{1/\alpha}$.

Which of the following pairs of functions are identical?

Let $R$ denote the set of all real numbers and $R^{+}$ denote the set of all positive real numbers. For the subsets $A$ and $B$ of $R$, define $f: A \rightarrow B$ by $f(x) = x^2$ for $x \in A$. Match the following lists:
| Column $I$ | Column $II$ |
| :--- | :--- |
| $A$. $f$ is one-one and onto, if | $1$. $A = R^{+}, B = R$ |
| $B$. $f$ is one-one but not onto, if | $2$. $A = B = R$ |
| $C$. $f$ is onto but not one-one, if | $3$. $A = R, B = R^{+}$ |
| $D$. $f$ is neither one-one nor onto, if | $4$. $A = B = R^{+}$ |

Let $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$. Then which of the following is true?

If $f(x) = \log_e \left( \frac{1-x}{1+x} \right)$,$|x| < 1$,then $f\left( \frac{2x}{1+x^2} \right)$ is equal to

If $f(x)$ is a real-valued function defined by $f(x) = \frac{a x^{10} + b x^8 + c x^6 + d x^4 + e x^2 + 12 x + 15}{x}$ for $x \neq 0$ and $f(4) = -4$, then find the value of $f(-4)$.