To find the domain of the function $f(x) = \frac{1}{1 - 7x}$, we need to determine the values of $x$ for which the function is defined.

Since $f(x)$ is a rational function, it is undefined where the denominator is zero. In this case, the denominator is $1 - 7x$. We need to find the value of $x$ that makes the denominator zero:

$1 - 7x = 0$

Solving for $x$, we get:

$7x = 1 \implies x = \frac{1}{7}$

Thus, the function $f(x) = \frac{1}{1 - 7x}$ is undefined when $x = \frac{1}{7}$. Therefore, the domain of $f(x)$ includes all real numbers except $x = \frac{1}{7}$.

Domain of $f(x)$:

$\boxed{(-\infty, \frac{1}{7}) \cup (\frac{1}{7}, \infty)}$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the domain of $g(x) = \frac{3}{2x + 5}$?
2. How do you find the domain of a function with both a square root and a fraction, like $h(x) = \frac{\sqrt{x+2}}{x-3}$?
3. What are the general rules for finding the domain of a logarithmic function, such as $k(x) = \log(x - 1)$?
4. How does the presence of an exponent in the denominator, such as $m(x) = \frac{1}{x^2 - 1}$, affect the domain?
5. What is the domain of the function $f(x) = \frac{1}{x^2 + 1}$?

Tip: Always set the denominator of a rational function not equal to zero to find the domain.