Discussion
Let f(x)=log5x(x>0) and g(x)=cos-1x(-1≤x≤1). Then the domain of g∘f is
Answer: D
The domain of \(g \circ f\) is \(\left[\frac{1}{5}, 5\right]\).
\(g(f(x)) = \cos^{-1}(\log_{5}x)\) needs \(x \gt 0\) for the logarithm and \(-1 \leq \log_{5}x \leq 1\) for the inverse cosine:
\[ -1 \leq \log_{5}x \leq 1 \iff 5^{-1} \leq x \leq 5^{1} \]
So the domain is \(\frac{1}{5} \leq x \leq 5\). The interval \([-1, 5]\) mixes the range condition with the answer and includes negative \(x\), where \(\log_{5}x\) is undefined. (0,1] and the half-lines starting at -1 or 0 (printed with \(\alpha\), evidently for infinity) ignore the restriction that the logarithm must lie between -1 and 1.
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