Question

Domain of the function $f\left(x\right)=\frac{3}{(4-x^2)}+{\log }_{10}\left(x^3-x\right)$ is

• A. $(1,2)$
• B. $(-1,0)\cup (1,2)$
• C. $(1,2)\cup (2,\infty )$
• D. $(-1,0)\cup (1,2)\cup (2,\infty )$

Answer 1

The correct option is D

$(-1,0)\cup (1,2)\cup (2,\infty )$

The explanation for correct option:

Simplify the given function part-wise.

Given: $f\left(x\right)=\frac{3}{(4-x^2)}+{\log }_{10}\left(x^3-x\right)$

Therefore,

$$\begin{gathered} (4-x^2)\ne 0 \\ \Rightarrow x^2\ne 4 \\ \Rightarrow x\ne \pm 2······\left(a\right) \end{gathered}$$

Again,

$$\begin{gathered} (x^3-x)>0 \\ \Rightarrow x(x+1)(x-1)>0 \\ \Rightarrow x>0,(x+1)>0,(x-1)>0 \\ \therefore x\in (-1,0)\cup (1,\infty )·····\left(b\right) \end{gathered}$$

Combining $\left(a\right)$ and $\left(b\right)$, we get:

$x\in (-1,0)\cup (1,2)\cup (2,\infty )$

Therefore, Option (D) is the correct answer.