The range of the real valued function $f(x) = \log_3(5 + 4x - x^2)$ is

$f(x)$ is a quadratic polynomial satisfying the condition $f(x) + f\left(\frac{1}{x}\right) = f(x) f\left(\frac{1}{x}\right)$. If $f(-1) = 0$,then the range of $f$ is

Let the sets $A$ and $B$ denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$ where $\lceil x \rceil$ denotes the smallest integer greater than or equal to $x$. Then among the statements
$(S1): A \cap B = (1, \infty) - \mathbb{N}$ and
$(S2): A \cup B = (1, \infty)$

The range of the function $f(x) = \begin{cases} 4x - 1, & x > 3 \\ x^2 - 2, & -2 \leq x \leq 3 \\ 3x + 4, & x < -2 \end{cases}$ is:

The domain of the function $f(x) = \log_{3+x}(x^2 - 1)$ is

The real-valued function $f(x) = \frac{\operatorname{cosec}^{-1} x}{\sqrt{x - [x]}}$,where $[x]$ denotes the greatest integer less than or equal to $x$,is defined for all $x$ belonging to: