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?

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.