Question

The domain of the real valued function \[ f(x)=\cos^{-1}\!\left(\log_{5}\frac{x}{5}\right)+\log_{5}\!\left(\cos^{-1}\frac{x}{5}\right) \] is:

• A. \(\left[\frac15,5\right]\)
• B. \([1,25]\)
• C. \(\left(\frac15,1\right)\)
• D. \([1,5)\)

Hint:

For logarithmic functions, always ensure the argument is strictly positive. For inverse trigonometric functions, the input must lie in the allowed interval.

Answer 1

The Correct Option is D

Concept: For the domain of a composite function, every individual expression must be defined.

• For \(\cos^{-1}(t)\), we need \[ -1\le t\le 1. \]
• For \(\log_5(t)\), we need \[ t>0. \]

Step 1: Apply the condition on the first term.
\[ \cos^{-1}\left(\log_5\frac{x}{5}\right) \] requires \[ -1\le \log_5\frac{x}{5}\le 1. \] Taking antilogarithm, \[ 5^{-1}\le \frac{x}{5}\le 5 \] \[ \frac15\le \frac{x}{5}\le 5 \] \[ 1\le x\le 25. \]

Step 2: Apply the condition on the second term.
\[ \log_5\left(\cos^{-1}\frac{x}{5}\right) \] requires \[ \cos^{-1}\frac{x}{5}>0. \] Also, \[ -1\le \frac{x}{5}\le 1. \] Hence \[ -5\le x\le 5. \] Now, \[ \cos^{-1}\frac{x}{5}>0 \] implies \[ \frac{x}{5}\neq 1 \] \[ x\neq 5. \] Therefore \[ -5\le x<5. \]

Step 3: Find the intersection.
\[ [1,25]\cap[-5,5) = [1,5). \] \[ \boxed{[1,5)} \] Hence, \[ \boxed{\text{Correct Option (4)}} \]