Let $f(x) = [x]^2 - [x+3] - 3, x \in \mathbb{R}$, where $[\bullet]$ is the greatest integer function. Then:

The function $f(x) = \text{sgn}(x) \cdot \sin x$ is

Let $f: X \rightarrow Y$ be a function and $A, B$ be non-void subsets of $Y$. Which of the following is true?

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 | $4$. $A = B = R^{+}$ |

The function $f(x) = x^{\frac{1}{\ln x}}$ is:

If $f(x) = 2\sin x$ and $g(x) = \cos^2 x$,then $(f + g)\left(\frac{\pi}{3}\right) = $