Let's analyze the given function $f(x) = \frac{x - 5}{x + 1}$ to determine its domain, x- and y-intercepts, and asymptotes.

1. Domain:

The domain of a rational function is all real numbers except where the denominator is zero.

The denominator is $x + 1$. Set the denominator equal to zero and solve for $x$: $x + 1 = 0 \implies x = -1$ So, the function is undefined at $x = -1$. Therefore, the domain is all real numbers except $-1$.

Domain in interval notation: $(-\infty, -1) \cup (-1, \infty)$

2. x-intercepts:

To find the x-intercepts, set $f(x) = 0$ and solve for $x$: $\frac{x - 5}{x + 1} = 0 \implies x - 5 = 0 \implies x = 5$ So, the x-intercept is $(5, 0)$.

3. y-intercepts:

To find the y-intercept, evaluate $f(x)$ at $x = 0$: $f(0) = \frac{0 - 5}{0 + 1} = \frac{-5}{1} = -5$ So, the y-intercept is $(0, -5)$.

4. Asymptotes:

• Vertical asymptote: The vertical asymptote occurs where the denominator is zero, i.e., $x = -1$.
• Horizontal asymptote: Since the degrees of the numerator and denominator are both 1 (linear functions), the horizontal asymptote is found by dividing the leading coefficients. The leading coefficients are both 1, so the horizontal asymptote is $y = 1$.

Summary:

• Domain: $(-\infty, -1) \cup (-1, \infty)$
• x-intercept: $(5, 0)$
• y-intercept: $(0, -5)$
• Vertical asymptote: $x = -1$
• Horizontal asymptote: $y = 1$

Would you like me to graph this function for you? Let me know if you have any questions or need more details.

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Here are 5 related questions to expand on this information:

1. How do you find slant asymptotes in a rational function?
2. What happens to the graph near the vertical asymptote?
3. How do you determine if a rational function has any holes in its graph?
4. Can the domain of a function ever include points where it is undefined?
5. How do you determine if a function has a removable discontinuity?

Tip: When solving rational functions, always check both the numerator and denominator for common factors before simplifying!