Key Concepts and Formulas

• Range of Sine Function: For any real number $\theta$, the value of $\sin \theta$ lies in the interval $[-1, 1]$. That is, $-1 \leq \sin \theta \leq 1$.
• Properties of Inequalities:
  • Multiplying or dividing an inequality by a negative number reverses the inequality signs.
  • Adding or subtracting a constant to all parts of an inequality does not change the inequality signs.
  • If $0 < a \leq X \leq b$, then $\frac{1}{b} \leq \frac{1}{X} \leq \frac{1}{a}$.

Step-by-Step Solution

We are asked to find the range of the function $f(x)=\frac{1}{7-\sin 5 x}$.

Step 1: Determine the range of $\sin 5x$. The argument of the sine function is $5x$. Since $x$ is defined on $\mathbf{R}$ (all real numbers), $5x$ can also take any real value. Therefore, the range of $\sin 5x$ is the same as the standard sine function: $-1 \leq \sin 5x \leq 1$

Step 2: Determine the range of $-\sin 5x$. To obtain the term $-\sin 5x$, we multiply the inequality from Step 1 by $-1$. When multiplying an inequality by a negative number, we must reverse the direction of the inequality signs: $(-1) \times (-1) \geq -\sin 5x \geq (-1) \times 1$ $1 \geq -\sin 5x \geq -1$ Rewriting this in the standard order (smallest value on the left): $-1 \leq -\sin 5x \leq 1$

Step 3: Determine the range of $7 - \sin 5x$. Now, we add $7$ to all parts of the inequality from Step 2. Adding a constant does not change the direction of the inequality signs: $7 + (-1) \leq 7 - \sin 5x \leq 7 + 1$ $6 \leq 7 - \sin 5x \leq 8$ This means the denominator of our function $f(x)$ lies in the interval $[6, 8]$.

Step 4: Determine the range of $f(x) = \frac{1}{7 - \sin 5x}$. We have established that $6 \leq 7 - \sin 5x \leq 8$. Since all values in this interval are positive ($6$ and $8$ are positive, and $7 - \sin 5x$ is always between them), we can take the reciprocal of all parts of the inequality. When taking the reciprocal of positive numbers in an inequality, the inequality signs are reversed: $\frac{1}{6} \geq \frac{1}{7 - \sin 5x} \geq \frac{1}{8}$ Rewriting this in the standard ascending order: $\frac{1}{8} \leq \frac{1}{7 - \sin 5x} \leq \frac{1}{6}$ Since $f(x) = \frac{1}{7 - \sin 5x}$, we have: $\frac{1}{8} \leq f(x) \leq \frac{1}{6}$

Step 5: State the range of $f(x)$. The inequality from Step 4 directly gives the range of the function $f(x)$. The range is the interval $\left[\frac{1}{8}, \frac{1}{6}\right]$.

Common Mistakes & Tips

• Inequality Reversal: Be extremely careful when multiplying or dividing inequalities by negative numbers, or when taking reciprocals of expressions that can be negative. Always reverse the inequality signs in these cases.
• Domain of Denominator: Ensure the denominator does not become zero or change sign within its range. If it did, the function would have asymptotes or require splitting the domain. Here, $7 - \sin 5x$ is always between $6$ and $8$, so it is always positive and non-zero.
• Step-by-Step Transformation: Build the expression inside the function systematically from the known range of the elementary function. For $f(x) = \frac{1}{g(x)}$, first find the range of $g(x)$, then transform it to find the range of $f(x)$.

Summary

The range of the function $f(x)=\frac{1}{7-\sin 5 x}$ is found by first determining the range of the sine term, then transforming this range to match the denominator $7-\sin 5x$, and finally taking the reciprocal. The known range of $\sin 5x$ is $[-1, 1]$. This leads to the range of $7-\sin 5x$ being $[6, 8]$. Taking the reciprocal of this positive interval gives the range of $f(x)$ as $\left[\frac{1}{8}, \frac{1}{6}\right]$.

The final answer is $\boxed{\text{(D)}}$.