Question

Let \( f(x)=\log_5 x \, (x > 0) \) and \( g(x)=\cos^{-1}(x) \, (-1\le x \le 1) \). Then the domain of \( g \circ f \) is

• A. \( (0,1] \)
• B. \( [-1,a) \)
• C. \( [0,a) \)
• D. \( \left[\frac{1}{5},5\right] \)
• E. \( [-1,5] \)

Hint:

For composite functions, always restrict the inner function so its output fits the domain of the outer function.

Answer 1

The Correct Option is D

Concept: For composition \( g(f(x)) \), the output of \( f(x) \) must lie in the domain of \( g(x) \).
• Domain of \( f(x)=\log_5 x \): \( x > 0 \)
• Domain of \( g(x)=\cos^{-1}x \): \( -1 \le x \le 1 \)

Step 1: Apply composition condition.
\[ -1 \le f(x) \le 1 \] \[ -1 \le \log_5 x \le 1 \]

Step 2: Convert into exponential form.
\[ \log_5 x \ge -1 \Rightarrow x \ge 5^{-1} = \frac{1}{5} \] \[ \log_5 x \le 1 \Rightarrow x \le 5^1 = 5 \]

Step 3: Combine with domain of \( f(x) \).
\[ \frac{1}{5} \le x \le 5 \]