Domain of the function $f(x)\,=\,\frac{1}{{\sqrt {(x + 1)({e^x} - 1)(x - 4)(x + 5)(x - 6)} }}$
$( - \infty , - 5) \cup ( - 1,4) \cup (6,8)$
$( - \infty , - 5) \cup ( - 1,0) \cup (0,4) \cup (6,\infty )$
$( - 5, - 1) \cup (0,4) \cup (6,\infty )$
$( - \infty , - 5) \cup ( - 1,4) \cup [6,\infty )$
If $f( x + y )=f( x ) f( y )$ and $\sum \limits_{ x =1}^{\infty} f( x )=2, x , y \in N$ where $N$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is
If the domain of the function $f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)$ is $[\alpha, \beta) \cup(\gamma, \delta]$, then $|3 \alpha+10(\beta+\gamma)+21 \delta|$ is equal to $.......$.
Product of all the solution of the equation ${x^{1 + {{\log }_{10}}x}} = 100000x$ is
The total number of functions,$f:\{1,2,3,4\} \cdot\{1,2,3,4,5,6\}$ such that $f (1)+ f (2)= f (3)$, is equal to .