https://file.aiquickdraw.com/mathbot/file/1725443243840r5m97ysq.jpg

The equation in the image is:

\[
y = \frac{x + 1}{x - 1}
\]

You are asked to find the **domain** and **range** of this rational function.

### Domain:
The domain of a function includes all the possible values of \( x \) for which the function is defined. For rational functions, the function is undefined when the denominator is zero because division by zero is not allowed.

Here, the denominator is \( x - 1 \). To find when the function is undefined:
\[
x - 1 = 0 \implies x = 1
\]
Thus, \( x = 1 \) is not in the domain.

**Domain**: \( x \in \mathbb{R} \) except \( x = 1 \).
This can be written as:
\[
\text{Domain: } (-\infty, 1) \cup (1, \infty)
\]

### Range:
To find the range of the function, we consider the possible values of \( y \). 

Start by solving for \( x \) in terms of \( y \):
\[
y = \frac{x + 1}{x - 1}
\]
Multiply both sides by \( x - 1 \):
\[
y(x - 1) = x + 1
\]
Expand and rearrange:
\[
yx - y = x + 1 \implies yx - x = y + 1
\]
Factor out \( x \):
\[
x(y - 1) = y + 1 \implies x = \frac{y + 1}{y - 1}
\]
The function is undefined when the denominator \( y - 1 = 0 \), which occurs when \( y = 1 \).

Thus, the function can take all real values of \( y \) except \( y = 1 \).

**Range**: \( y \in \mathbb{R} \) except \( y = 1 \).
This can be written as:
\[
\text{Range: } (-\infty, 1) \cup (1, \infty)
\]

Would you like more details or have any further questions?

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Here are 5 related questions you might find interesting:
1. How do you find the vertical and horizontal asymptotes of the function \( y = \frac{x + 1}{x - 1} \)?
2. What is the behavior of the function \( y = \frac{x + 1}{x - 1} \) as \( x \) approaches 1 from the left and right?
3. How do you determine the intercepts of the function \( y = \frac{x + 1}{x - 1} \)?
4. Can you describe the transformations of the parent function \( y = \frac{1}{x} \) that result in \( y = \frac{x + 1}{x - 1} \)?
5. How would you sketch the graph of \( y = \frac{x + 1}{x - 1} \) based on the domain, range, and asymptotes?

**Tip:** When finding the range of a rational function, solving for \( x \) in terms of \( y \) can often help identify the values that \( y \) cannot take.

Find the domain and range of the rational function y = (x + 1) / (x - 1).

Understand how to find the domain and range of the rational function y = (x + 1) / (x - 1). This problem involves core algebra concepts such as domain, range, and rational functions, suitable for students in Grades 9-10.

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