The given function is:

$y = \frac{7}{3x + 7}$

To find the domain, we need to identify any restrictions on the values of $x$ that would cause the function to be undefined. Since this is a rational function (a fraction), the only restriction comes from the denominator. The denominator cannot be zero, because division by zero is undefined.

Thus, we need to find when the denominator equals zero:

$3x + 7 = 0$

Solve for $x$:

$3x = -7$

$x = -\frac{7}{3}$

Therefore, the function is undefined when $x = -\frac{7}{3}$.

Thus, the domain of the function is all real numbers except $x = -\frac{7}{3}$, or in interval notation:

$\boxed{(-\infty, -\frac{7}{3}) \cup (-\frac{7}{3}, \infty)}$

Would you like more details or further clarifications?

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Here are 5 related questions for further exploration:

1. What happens to the graph of $y = \frac{7}{3x + 7}$ as $x$ approaches $-\frac{7}{3}$?
2. How do you find the domain of a rational function with a quadratic denominator?
3. What is the range of the function $y = \frac{7}{3x + 7}$?
4. How do you determine asymptotes for a rational function?
5. How do transformations (such as shifts and stretches) affect the domain of a function?

Tip: Always remember that the domain of a rational function is determined by setting the denominator equal to zero and solving for $x$. Any $x$-values that make the denominator zero are excluded from the domain.