If y = f(x) = `(x+2)/(x-1) , x ne1` , then show that x = f(y) .

Explanation

To show that \( x = f(y) \) given \( y = f(x) = \frac{x+2}{x-1} \), we will follow these steps:

### Step 1: Express \( f(y) \)
We start by substituting \( y \) into the function \( f \). Since \( y = f(x) \), we have:
\[ f(y) = f\left(\frac{x+2}{x-1}\right) \]
Now, we replace \( x \) in the function \( f(x) = \frac{x+2}{x-1} \) with \( y \):
\[ f(y) = \frac{y + 2}{y - 1} \]

### Step 2: Substitute \( y \) back into \( f(y) \)
Since \( y = \frac{x+2}{x-1} \), we substitute this into our expression for \( f(y) \):
\[ f(y) = \frac{\frac{x+2}{x-1} + 2}{\frac{x+2}{x-1} - 1} \]

### Step 3: Simplify the numerator
The numerator becomes:
\[ \frac{x+2}{x-1} + 2 = \frac{x+2 + 2(x-1)}{x-1} = \frac{x+2 + 2x - 2}{x-1} = \frac{3x}{x-1} \]

### Step 4: Simplify the denominator
The denominator becomes:
\[ \frac{x+2}{x-1} - 1 = \frac{x+2 - (x-1)}{x-1} = \frac{x+2 - x + 1}{x-1} = \frac{3}{x-1} \]

### Step 5: Combine the simplified numerator and denominator
Now we can combine the simplified numerator and denominator:
\[ f(y) = \frac{\frac{3x}{x-1}}{\frac{3}{x-1}} = \frac{3x}{3} = x \]

### Conclusion
Thus, we have shown that:
\[ f(y) = x \]
which implies that:
\[ x = f(y) \]