To ascertain the domain of \( f(x) = \cos^{-1}(7x) \), we must first establish the valid input range for the inverse cosine function. The inverse cosine function, \( \cos^{-1}(x) \), accepts inputs only from the interval \([-1, 1]\).
Consequently, for \( f(x) = \cos^{-1}(7x) \) to be defined, the expression \( 7x \) must fall within this interval: \(-1 \leq 7x \leq 1\).
Solving these inequalities yields:
1. \(-1 \leq 7x\)
Dividing by 7 gives:
\(-\frac{1}{7} \leq x\)
2. \(7x \leq 1\)
Dividing by 7 again gives:
\(x \leq \frac{1}{7}\)
Combining these results, we establish the domain as:
\(-\frac{1}{7} \leq x \leq \frac{1}{7}\)
Thus, the domain of the function \( f(x) = \cos^{-1}(7x) \) is \( \left[ -\frac{1}{7}, \frac{1}{7} \right] \).