If `A=R-{b}` and `B=R-{1}` and function `f:AtoB` is defined by `f(x)=(x-a)/(x-b),aneb`, then `f^(-1)(x)` is

Options

A. `(bx-a)/(x-1)`
B. `(a=bx)/(x-1)`
C. `(ax-b)/(x-1)`
D. none of these

Answer

To find the inverse of the function \( f: A \to B \) defined by \( f(x) = \frac{x - a}{x - b} \), we will follow these steps:

### Step 1: Set up the equation
Let \( f(x) = y \). Therefore, we have:
\[ y = \frac{x - a}{x - b} \]

### Step 2: Cross-multiply
To eliminate the fraction, we cross-multiply:
\[ y(x - b) = x - a \]

### Step 3: Expand the equation
Expanding the left side gives:
\[ yx - by = x - a \]

### Step 4: Rearrange the equation
Rearranging the equation to isolate terms involving \( x \):
\[ yx - x = by - a \]
Factoring out \( x \) from the left side:
\[ x(y - 1) = by - a \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{by - a}{y - 1} \]

### Step 6: Write the inverse function
Since we set \( y = f(x) \), we can express the inverse function \( f^{-1}(x) \):
\[ f^{-1}(x) = \frac{bx - a}{x - 1} \]

Thus, the final answer is:
\[ f^{-1}(x) = \frac{bx - a}{x - 1} \]