# The domain of the definition

Question:

The domain of the definition of the function $f(x)=\frac{1}{4-x^{2}}+\log _{10}\left(x^{3}-x\right)$ is:

1. (1) $(-1,0) \cup(1,2) \cup(3, \infty)$

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

3. (3) $(-1,0) \cup(1,2) \cup(2, \infty)$

4. (4) $(1,2) \cup(2, \infty)$

Correct Option: , 3

Solution:

To determine domain, denominator $\neq 0$ and $x^{3}-x>0$

i.e., $4-x^{2} \neq 0 x \neq \pm 2$ .......(1)

and $x(x-1)(x+1)>0$

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

Hence domain is intersection of $(1) \&(2)$.

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