If `f(x) =(ax-b)/(bx-a)=y`
Find the value of f(y).

Answer

To find the value of \( f(y) \) given that \( f(x) = \frac{ax - b}{bx - a} = y \), we will follow these steps:

### Step 1: Substitute \( y \) into the function
We start by substituting \( y \) into the function \( f \):
\[ f(y) = f\left(\frac{ax - b}{bx - a}\right) \]

### Step 2: Replace \( x \) in the function
Now we replace \( x \) in the function with \( \frac{ax - b}{bx - a} \):
\[ f\left(\frac{ax - b}{bx - a}\right) = \frac{a\left(\frac{ax - b}{bx - a}\right) - b}{b\left(\frac{ax - b}{bx - a}\right) - a} \]

### Step 3: Simplify the numerator
Let's simplify the numerator:
\[ a\left(\frac{ax - b}{bx - a}\right) - b = \frac{a(ax - b) - b(bx - a)}{bx - a} \]
Expanding this gives:
\[ = \frac{a^2x - ab - b^2x + ab}{bx - a} = \frac{(a^2 - b^2)x}{bx - a} \]

### Step 4: Simplify the denominator
Now simplify the denominator:
\[ b\left(\frac{ax - b}{bx - a}\right) - a = \frac{b(ax - b) - a(bx - a)}{bx - a} \]
Expanding this gives:
\[ = \frac{abx - b^2 - abx + a^2}{bx - a} = \frac{a^2 - b^2}{bx - a} \]

### Step 5: Combine the results
Now we can combine the results from the numerator and denominator:
\[ f(y) = \frac{\frac{(a^2 - b^2)x}{bx - a}}{\frac{a^2 - b^2}{bx - a}} = x \]

### Conclusion
Thus, we find that:
\[ f(y) = x \]