https://file.aiquickdraw.com/mathbot/file/1726846851455tpexok9s.jpg

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.

What is the domain of f(x) = x / (x - 7)?

Find the domain of the rational function f(x) = x / (x - 7). The solution covers algebraic concepts related to rational functions, focusing on identifying when the denominator is zero. Suitable for Grades 9-12.

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