The function in the image is given as:

$f(x) = \frac{x}{x - 7}$

To find the domain of $f(x)$, we need to identify the values of $x$ for which the function is defined. The function is undefined when the denominator is equal to zero. So, we must solve the equation:

$x - 7 = 0$

Solving for $x$, we get:

$x = 7$

This means the function is undefined at $x = 7$. Therefore, the domain of $f(x)$ is all real values of $x$ except 7.

Thus, the correct answer is:

$\text{C: All real values of } x \text{ such that } x \neq 7$

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How would the domain change if the function was $f(x) = \frac{x+2}{x-7}$?
2. What happens to the domain if the denominator involves a quadratic expression?
3. Can a function have more than one point where it is undefined? If so, how do you handle such cases?
4. How do you express the domain of a function using interval notation?
5. What happens to the range of the function when a certain value of $x$ is excluded from the domain?

Tip: Always check for values that make the denominator zero when determining the domain of a rational function.