To solve the equation \( f(x) = \frac{x + 1}{x - 5} \) for \( f(x) = 0 \), we need to find the values of \( x \) that make the function equal to zero.

\[
\frac{x + 1}{x - 5} = 0
\]

For a fraction to be zero, the numerator must be zero while the denominator is non-zero. Thus, we solve the equation:

\[
x + 1 = 0
\]

\[
x = -1
\]

So, the solution is \( x = -1 \).

**Note:** The solution is valid as long as the denominator \( x - 5 \) is not zero. In this case, \( x = 5 \) would make the denominator zero, but it is not part of the solution because it would make the original function undefined.

Would you like more details on this, or do you have any other questions?

Here are 5 related questions:

1. How can we determine where \( f(x) \) is undefined?
2. What is the significance of the vertical asymptote in the context of this function?
3. How do you determine if the function \( f(x) \) is increasing or decreasing?
4. How do we find the inverse of the function \( f(x) \)?
5. What is the behavior of \( f(x) \) as \( x \) approaches the vertical asymptote \( x = 5 \)?

**Tip:** When solving rational equations, always check that the solution doesn't make any denominators zero, as this would make the original equation undefined.

Learn how to solve the equation f(x) = (x + 1) / (x - 5) for f(x) = 0 with a detailed step-by-step solution. This problem covers rational functions, equations, and function behavior. Suitable for high school students.

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