Question

Let $f(x) = \frac{1}{7 - \sin 5x}$ be a function defined on $\mathbb{R}$. Then the range of the function $f(x)$ is equal to:

• A. $\left[\frac{1}{8}, \frac{1}{5}\right]$
• B. $\left[\frac{1}{7}, \frac{1}{6}\right]$
• C. $\left[\frac{1}{7}, \frac{1}{5}\right]$
• D. $\left[\frac{1}{8}, \frac{1}{6}\right]$

Answer 1

The Correct Option is D

Given that \( \sin 5x \in [-1, 1] \), it follows that:
\[-\sin 5x \in [-1, 1]\]
Consequently:
\[7 - \sin 5x \in [6, 8]\]
This implies that the function \( f(x) = \frac{1}{7 - \sin 5x} \) yields values within the interval:
\[\frac{1}{7 - \sin 5x} \in \left[\frac{1}{8}, \frac{1}{6}\right]\]
Thus, the range of \( f(x) \) is determined to be:
\[\left[\frac{1}{8}, \frac{1}{6}\right].\]