Question

The domain of \( f(x) = \cos^{-1}(7x) \) is:

• A. \( \left[ -\frac{1}{7}, \frac{1}{7} \right] \)
• B. \( [-7, 7] \)
• C. \( [0, 7] \)
• D. \( [-1, 1] \)

Answer 1

The Correct Option is A

To determine the domain of the function \( f(x) = \cos^{-1}(7x) \), we must first understand the constraints of the inverse cosine function. The inverse cosine function, \( \cos^{-1}(x) \), is defined only for values of \( x \) in the interval \([-1, 1]\). 

For \( f(x) = \cos^{-1}(7x) \) to be valid, the expression \( 7x \) must also lie within the domain of the inverse cosine function: \(-1 \leq 7x \leq 1\).

Let us solve the inequalities:

1. \(-1 \leq 7x\)

Divide both sides by 7:

\(-\frac{1}{7} \leq x\)

2. \(7x \leq 1\)

Again, divide both sides by 7:

\(x \leq \frac{1}{7}\)

Combining these two inequalities, we find:

\(-\frac{1}{7} \leq x \leq \frac{1}{7}\)

Therefore, the domain of the function \( f(x) = \cos^{-1}(7x) \) is \( \left[ -\frac{1}{7}, \frac{1}{7} \right] \).

Answer 2

The function \( \cos^{-1}(x) \) is defined only for \( x \in [-1, 1] \). Here, \( f(x) = \cos^{-1}(7x) \), so \( 7x \) must also lie in the interval \([-1, 1]\). Solve the inequality:

\(-1 \leq 7x \leq 1\).

Divide through by 7:

\(-\frac{1}{7} \leq x \leq \frac{1}{7}\).

Thus, the domain of \( f(x) = \cos^{-1}(7x) \) is \(\left[ -\frac{1}{7}, \frac{1}{7} \right]\).