Question:
The domain of the real valued function $f(x) = \sin\left(\log\left(\frac{\sqrt{4 - x^2}}{1 - x}\right)\right)$ is
The domain of the real valued function $f(x) = \sin\left(\log\left(\frac{\sqrt{4 - x^2}}{1 - x}\right)\right)$ is
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Always check domain conditions for logarithm and square root individually and combine them logically.
Updated On: May 19, 2025
- $(1, 4)$
- $(-1, 1)$
- $(-2, 1)$
- $(-2, 4)$
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The Correct Option is C
Solution and Explanation
To find the domain, the argument of the logarithm must be positive:
$\frac{\sqrt{4 - x^2}}{1 - x}>0$
We need $4 - x^2 \geq 0 \Rightarrow x \in [-2, 2]$ and $1 - x>0 \Rightarrow x<1$
So, domain is $x \in (-2, 1)$
$\frac{\sqrt{4 - x^2}}{1 - x}>0$
We need $4 - x^2 \geq 0 \Rightarrow x \in [-2, 2]$ and $1 - x>0 \Rightarrow x<1$
So, domain is $x \in (-2, 1)$
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