The domain of definition of the function f(x) | vedclass

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Question

(D) For the function $f(x) = \frac{3}{4 - x^2} + \log_{10}(x^3 - x)$ to be defined:$1$. The denominator must not be zero: $4 - x^2 
eq 0$ $\Rightarro

Vedclass · Accepted Answer

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

Vedclass · Answer

(D) For the function $f(x) = \frac{3}{4 - x^2} + \log_{10}(x^3 - x)$ to be defined:$1$. The denominator must not be zero: $4 - x^2 
eq 0$ $\Rightarrow x^2 
eq 4$ $\Rightarrow x 
eq \pm 2$.$2$. The argument of the logarithm must be positive: $x^3 - x > 0$.Factoring the inequality: $x(x^2 - 1) > 0 \Rightarrow x(x - 1)(x + 1) > 0$.Using the wavy curve method (sign scheme) on the number line with critical points $-1, 0, 1$,the expression is positive in the intervals $(-1, 0) \cup (1, \infty)$.Combining the conditions: $x \in (-1, 0) \cup (1, \infty)$ and $x 
eq \pm 2$.Since $2$ lies in the interval $(1, \infty)$,we must exclude it.Thus,the domain is $D = (-1, 0) \cup (1, 2) \cup (2, \infty)$.

The domain of definition of the function $f(x) = \frac{3}{4 - x^2} + \log_{10}(x^3 - x)$ is

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