If `f(x)=(x)/(x-1)`, then evaluate : `(f(a//b))/(f(b//a))`

Answer

To evaluate \(\frac{f\left(\frac{a}{b}\right)}{f\left(\frac{b}{a}\right)}\) where \(f(x) = \frac{x}{x-1}\), we will follow these steps:

### Step 1: Calculate \(f\left(\frac{a}{b}\right)\)

Using the function definition:
\[ f\left(\frac{a}{b}\right) = \frac{\frac{a}{b}}{\frac{a}{b} - 1} \]

Now, simplify the denominator:
\[ \frac{a}{b} - 1 = \frac{a}{b} - \frac{b}{b} = \frac{a - b}{b} \]

Thus, we have:
\[ f\left(\frac{a}{b}\right) = \frac{\frac{a}{b}}{\frac{a-b}{b}} = \frac{a}{a-b} \]

### Step 2: Calculate \(f\left(\frac{b}{a}\right)\)

Now, we calculate:
\[ f\left(\frac{b}{a}\right) = \frac{\frac{b}{a}}{\frac{b}{a} - 1} \]

Again, simplify the denominator:
\[ \frac{b}{a} - 1 = \frac{b}{a} - \frac{a}{a} = \frac{b - a}{a} \]

Thus, we have:
\[ f\left(\frac{b}{a}\right) = \frac{\frac{b}{a}}{\frac{b-a}{a}} = \frac{b}{b-a} \]

### Step 3: Calculate \(\frac{f\left(\frac{a}{b}\right)}{f\left(\frac{b}{a}\right)}\)

Now we substitute the values we found into the expression:
\[ \frac{f\left(\frac{a}{b}\right)}{f\left(\frac{b}{a}\right)} = \frac{\frac{a}{a-b}}{\frac{b}{b-a}} \]

This can be simplified as:
\[ = \frac{a}{a-b} \cdot \frac{b-a}{b} = \frac{a(b-a)}{b(a-b)} \]

Since \(b-a = -(a-b)\), we can rewrite this as:
\[ = \frac{a(-1)(a-b)}{b(a-b)} = -\frac{a}{b} \]

### Final Result

Thus, the final result is:
\[ \frac{f\left(\frac{a}{b}\right)}{f\left(\frac{b}{a}\right)} = -\frac{a}{b} \]

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