Functions 1 Question 1
1. The domain of the definition of the function $f(x)=\frac{1}{4-x^{2}}+\log _{10}\left(x^{3}-x\right)$ is
(2019 Main, 9 April II)
(a) $(-1,0) \cup(1,2) \cup(3, \infty)$
(b) $(-2,-1) \cup(-1,0) \cup(2, \infty)$
(c) $(-1,0) \cup(1,2) \cup(2, \infty)$
(d) $(1,2) \cup(2, \infty)$
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Answer:
Correct Answer: 1. (c)
Solution:
- Given function $f(x)=\frac{1}{4-x^{2}}+\log _{10}\left(x^{3}-x\right)$
For domain of $f(x)$
$$ \begin{array}{lrl} & & 4-x^{2} \neq 0 \Rightarrow x \neq \pm 2 \\ \text { and } & & x^{3}-x>0 \\ \Rightarrow & & x(x-1)(x+1)>0 \end{array} $$
From Wavy curve method,
$x \in(-1,0) \cup(1, \infty)$
From Eqs. (i) and (ii), we get the domain of $f(x)$ as $(-1,0) \cup(1,2) \cup(2, \infty)$.
