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

Options

A. one-one but not onto
B. onto but not one-one
C. both one-one and onto
D. neither one-one nor onto

Answer

To solve the problem, we need to analyze the function \( f: A \to B \) defined by

\[ f(x) = \frac{x - a}{x - b} \]

where \( A = \mathbb{R} - \{b\} \) and \( B = \mathbb{R} - \{1\} \). We need to find the inverse of the function \( f \).

### Step 1: Set \( f(x) = y \)

Let \( y = f(x) \). Therefore, we have:

\[ y = \frac{x - a}{x - b} \]

### Step 2: Cross-multiply to eliminate the fraction

Cross-multiplying gives us:

\[ y(x - b) = x - a \]

### Step 3: Distribute \( y \)

Distributing \( y \) on the left side results in:

\[ yx - by = x - a \]

### Step 4: Rearrange the equation to isolate \( x \)

Now, we want to isolate \( x \). Rearranging the equation gives us:

\[ yx - x = by - a \]

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 have expressed \( x \) in terms of \( y \), we can write the inverse function \( f^{-1}(y) \):

\[ f^{-1}(y) = \frac{by - a}{y - 1} \]

### Conclusion

Thus, the inverse function \( f^{-1}(x) \) is:

\[ f^{-1}(x) = \frac{bx - a}{x - 1} \]