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

Answer: (D)

$\left[\frac{1}{8}, \frac{1}{6}\right]$

Answer 2

Since $\sin 5x \in [-1, 1]$, we have:
$-\sin 5x \in [-1, 1]$
Therefore:
$7 - \sin 5x \in [6, 8]$
Thus, the function $f(x) = \frac{1}{7 - \sin 5x}$ takes values in the interval:
$\frac{1}{7 - \sin 5x} \in \left[\frac{1}{8}, \frac{1}{6}\right]$
Therefore, the range of $f(x)$ is:
$\left[\frac{1}{8}, \frac{1}{6}\right].$