Find the inverse of each relation and the domain and range 4. Y = 2x over x-1

To find the inverse of the relation \( y = \frac{2x}{x - 1} \), follow these steps:

### Step 1: Write the equation in terms of \( x \) and \( y \)
\[
y = \frac{2x}{x - 1}
\]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, swap \( x \) and \( y \):
\[
x = \frac{2y}{y - 1}
\]

### Step 3: Solve for \( y \)
Now, solve this equation for \( y \):

Multiply both sides by \( y - 1 \) to eliminate the denominator:
\[
x(y - 1) = 2y
\]

Expand the left-hand side:
\[
xy - x = 2y
\]

Move all terms involving \( y \) to one side:
\[
xy - 2y = x
\]

Factor out \( y \) from the left-hand side:
\[
y(x - 2) = x
\]

Now, solve for \( y \):
\[
y = \frac{x}{x - 2}
\]

### Step 4: The inverse relation
The inverse of the relation is:
\[
y^{-1} = \frac{x}{x - 2}
\]

### Domain and Range of the Inverse

#### Original Function:
- **Domain**: The function \( y = \frac{2x}{x - 1} \) is undefined when the denominator is zero, which happens when \( x = 1 \). Therefore, the domain is all real numbers except \( x = 1 \):
  \[
  \text{Domain: } (-\infty, 1) \cup (1, \infty)
  \]
  
- **Range**: Since the function has a vertical asymptote at \( x = 1 \), the range excludes the value the function cannot output. By analyzing the behavior, we find that \( y \neq 2 \). Thus, the range is:
  \[
  \text{Range: } (-\infty, 2) \cup (2, \infty)
  \]

#### Inverse Function:
- **Domain**: The inverse function \( y = \frac{x}{x - 2} \) is undefined when \( x = 2 \) (since the denominator is zero at \( x = 2 \)). Therefore, the domain of the inverse is:
  \[
  \text{Domain: } (-\infty, 2) \cup (2, \infty)
  \]
  
- **Range**: The range of the inverse function corresponds to the domain of the original function. Therefore, the range is:
  \[
  \text{Range: } (-\infty, 1) \cup (1, \infty)
  \]

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Would you like more details on any step, or do you have further questions?

Here are 5 related questions you might explore:
1. How do you graph the original function and its inverse?
2. What happens to the domain and range if the function is modified to \( y = \frac{2x + 1}{x - 1} \)?
3. How do you confirm that two functions are inverses?
4. What are asymptotes, and how do they affect the behavior of functions?
5. How do we find the inverse of more complex rational functions?

**Tip:** When finding the inverse, always check if the function is one-to-one. Functions that aren’t one-to-one don't have inverses without restrictions.

Find the inverse of each relation and the domain and range for Y = 2x over x - 1.

Discover how to find the inverse of the rational function y = 2x / (x - 1), including steps for solving the equation, determining the domain and range of both the original and inverse functions. This solution covers key algebraic concepts, suitable for Grades 10-12 students.

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