 List the domain and the​ x- and​ y-intercepts of the following function. Graph the function. Be sure to label all the asymptotes.

​f(x)equals=StartFraction x minus 5 Over x plus 1 EndFractionx−5x+1 

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Part 1

What is the domain of the​ function?

enter your response here 

​(Type your answer in interval​ notation.)




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!

List the domain and the x- and y-intercepts of the following function. Graph the function. Be sure to label all the asymptotes.
f(x) = (x - 5) / (x + 1)

What is the domain of the function? (Type your answer in interval notation.)

Learn how to find the domain, x- and y-intercepts, and asymptotes of the rational function f(x) = (x - 5) / (x + 1). This problem explores rational function properties, including vertical and horizontal asymptotes, with step-by-step solutions for high school students. Suitable for Grades 10-12.

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