If `f(x)=(x-1)/(x+1)`, then f(ax) in terms of f(x) is equal to

Options

A. `(f(x)+a)/(1+af(x))`
B. `((a-1) f(x)+a+1)/((a+1)f(x)+a-1)`
C. `((a+1)f(x)+a-1)/((a-1)f(x) +a+1)`
D. None of these

Answer

To find \( f(ax) \) in terms of \( f(x) \), where \( f(x) = \frac{x-1}{x+1} \), we will follow these steps:

### Step 1: Write the expression for \( f(ax) \)
We start by substituting \( ax \) into the function \( f \):
\[ f(ax) = \frac{ax - 1}{ax + 1} \]

### Step 2: Express \( x \) in terms of \( f(x) \)
From the original function, we can express \( x \) in terms of \( f(x) \):
\[ f(x) = \frac{x-1}{x+1} \]
Cross-multiplying gives:
\[ f(x)(x + 1) = x - 1 \]
Rearranging this, we get:
\[ f(x)x + f(x) = x - 1 \implies f(x)x - x = -1 - f(x) \implies x(f(x) - 1) = -1 - f(x) \]
Thus,
\[ x = \frac{-1 - f(x)}{f(x) - 1} \]

### Step 3: Substitute \( x \) back into \( f(ax) \)
Now we substitute this expression for \( x \) back into our expression for \( f(ax) \):
\[ f(ax) = \frac{a\left(\frac{-1 - f(x)}{f(x) - 1}\right) - 1}{a\left(\frac{-1 - f(x)}{f(x) - 1}\right) + 1} \]

### Step 4: Simplify the expression
Let's simplify the numerator and denominator separately.

**Numerator:**
\[ a\left(\frac{-1 - f(x)}{f(x) - 1}\right) - 1 = \frac{-a(1 + f(x)) - (f(x) - 1)}{f(x) - 1} = \frac{-a - af(x) - f(x) + 1}{f(x) - 1} \]
This simplifies to:
\[ \frac{1 - (a + 1)f(x)}{f(x) - 1} \]

**Denominator:**
\[ a\left(\frac{-1 - f(x)}{f(x) - 1}\right) + 1 = \frac{-a(1 + f(x)) + (f(x) - 1)}{f(x) - 1} = \frac{-a - af(x) + f(x) - 1}{f(x) - 1} \]
This simplifies to:
\[ \frac{(1 - a) + (1 - a)f(x)}{f(x) - 1} \]

### Step 5: Combine and simplify
Putting it all together:
\[ f(ax) = \frac{1 - (a + 1)f(x)}{(1 - a) + (1 - a)f(x)} \]
This can be further simplified to:
\[ f(ax) = \frac{(1 - (a + 1)f(x))}{(1 - a)(1 + f(x))} \]

### Step 6: Final expression
Thus, we can express \( f(ax) \) in terms of \( f(x) \):
\[ f(ax) = \frac{(1 - (a + 1)f(x))}{(1 - a)(1 + f(x))} \]

### Conclusion
The correct option from the given choices is:
\[ \text{Option C: } f(ax) = \frac{(a - 1)f(x) + (a + 1)}{(a + 1)f(x) + (a - 1)} \]