Question

If $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\sqrt{(\mathrm{x}+1)({\mathrm{e}}^{\mathrm{x}}-1)(\mathrm{x}-4)(\mathrm{x}+5)(\mathrm{x}-6)}}$, then the domain of $f\left(x\right)$ is

• A. (-∞,-5) ∪ (-1,4) ∪ (6,8)
• B. (-∞,-5) ∪ (-1,0) ∪ (0,4) ∪ (6,∞)
• C. (-5,1) ∪ (0,4) ∪ (6,∞)
• D. None of these

Answer 1

The correct option is C

(-5,1) ∪ (0,4) ∪ (6,∞)

Step 1: Given information :

$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\sqrt{(\mathrm{x}+1)({\mathrm{e}}^{\mathrm{x}}-1)(\mathrm{x}-4)(\mathrm{x}+5)(\mathrm{x}-6)}}$

Step 2: Determination of the above equation :

For $f\left(x\right)$ to exist,

$$\begin{gathered} (\mathrm{x}+1)({\mathrm{e}}^{\mathrm{x}}-1)(\mathrm{x}-4)(\mathrm{x}+5)(\mathrm{x}-6)>0 \\ \Rightarrow \mathrm{x}\in (-5,-1)\cup (0,4)\cup (6,\infty ) \end{gathered}$$

Hence, the correct option is option (c).