If `f(x) = (( x + 1))/( ( x - 1))` , then the ratio of x to f(y) where ` y = f(x)` is

Options

A. x : y
B. `x^(2) : y^(2)`
C. `1 : 1 `
D. ` y : x`

Answer

To solve the problem step by step, we need to find the ratio of \( x \) to \( f(y) \) where \( y = f(x) \) and \( f(x) = \frac{x + 1}{x - 1} \).

### Step 1: Find \( f(y) \)
Given \( f(x) = \frac{x + 1}{x - 1} \), we first express \( f(y) \) in terms of \( y \):
\[ f(y) = \frac{y + 1}{y - 1} \]

### Step 2: Substitute \( y \) with \( f(x) \)
Since \( y = f(x) \), we substitute \( f(x) \) into the equation:
\[ y = \frac{x + 1}{x - 1} \]
Now we can express \( f(y) \) as:
\[ f(y) = f\left(\frac{x + 1}{x - 1}\right) = \frac{\left(\frac{x + 1}{x - 1}\right) + 1}{\left(\frac{x + 1}{x - 1}\right) - 1} \]

### Step 3: Simplify \( f(y) \)
Now we simplify \( f(y) \):
1. The numerator:
\[ \frac{x + 1}{x - 1} + 1 = \frac{x + 1 + (x - 1)}{x - 1} = \frac{2x}{x - 1} \]
2. The denominator:
\[ \frac{x + 1}{x - 1} - 1 = \frac{x + 1 - (x - 1)}{x - 1} = \frac{2}{x - 1} \]
Thus, we have:
\[ f(y) = \frac{\frac{2x}{x - 1}}{\frac{2}{x - 1}} = x \]

### Step 4: Find the ratio of \( x \) to \( f(y) \)
Now we find the ratio of \( x \) to \( f(y) \):
\[ \text{Ratio} = \frac{x}{f(y)} = \frac{x}{x} \]
Since \( f(y) = x \), we have:
\[ \frac{x}{x} = 1 \]

### Final Answer
Thus, the ratio of \( x \) to \( f(y) \) is:
\[ \text{Ratio} = 1 : 1 \]

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