find inverse of the function f(x)=2x-3/x-1

To find the inverse of the function \( f(x) = \frac{2x - 3}{x - 1} \), follow these steps:

### Step 1: Set \( f(x) = y \)
We begin by writing the equation as:
\[
y = \frac{2x - 3}{x - 1}
\]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, we switch \( x \) and \( y \) in the equation:
\[
x = \frac{2y - 3}{y - 1}
\]

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

1. Multiply both sides by \( y - 1 \) to eliminate the denominator:
\[
x(y - 1) = 2y - 3
\]
\[
xy - x = 2y - 3
\]

2. Rearrange the equation to get all terms involving \( y \) on one side:
\[
xy - 2y = x - 3
\]

3. Factor out \( y \) from the left side:
\[
y(x - 2) = x - 3
\]

4. Solve for \( y \):
\[
y = \frac{x - 3}{x - 2}
\]

### Step 4: Write the inverse function
The inverse function is:
\[
f^{-1}(x) = \frac{x - 3}{x - 2}
\]

Would you like more details or clarifications?

Here are 5 related questions to explore:
1. What are the domain and range of the original function \( f(x) \)?
2. How do we verify that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \)?
3. What is the domain and range of the inverse function \( f^{-1}(x) \)?
4. How does the vertical asymptote of \( f(x) \) affect its inverse?
5. What happens to the graph of a function when you take its inverse?

**Tip**: Always check if the inverse function exists by confirming that the original function is one-to-one (i.e., passes the horizontal line test).

Learn how to find the inverse of the function f(x) = (2x - 3) / (x - 1) using algebraic manipulation. This problem covers key algebraic concepts such as inverse functions and solving equations, making it suitable for students in Grades 10-12.

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