If `f(x)=(3x-1)/(x+1),xne-1`, then find fof (x).

Answer

To find \( f(f(x)) \) where \( f(x) = \frac{3x - 1}{x + 1} \) and \( x \neq -1 \), we will follow these steps:

### Step 1: Substitute \( f(x) \) into itself
We start by substituting \( f(x) \) into the function itself:
\[ f(f(x)) = f\left(\frac{3x - 1}{x + 1}\right) \]

### Step 2: Replace \( x \) in \( f(x) \)
Now, we will replace \( x \) in the function \( f(x) \) with \( \frac{3x - 1}{x + 1} \):
\[ f\left(\frac{3x - 1}{x + 1}\right) = \frac{3\left(\frac{3x - 1}{x + 1}\right) - 1}{\left(\frac{3x - 1}{x + 1}\right) + 1} \]

### Step 3: Simplify the numerator
Let's simplify the numerator:
\[ 3\left(\frac{3x - 1}{x + 1}\right) - 1 = \frac{9x - 3}{x + 1} - 1 = \frac{9x - 3 - (x + 1)}{x + 1} = \frac{9x - 3 - x - 1}{x + 1} = \frac{8x - 4}{x + 1} \]

### Step 4: Simplify the denominator
Now, simplify the denominator:
\[ \left(\frac{3x - 1}{x + 1}\right) + 1 = \frac{3x - 1}{x + 1} + \frac{x + 1}{x + 1} = \frac{3x - 1 + x + 1}{x + 1} = \frac{4x}{x + 1} \]

### Step 5: Combine the results
Now we can combine the simplified numerator and denominator:
\[ f(f(x)) = \frac{\frac{8x - 4}{x + 1}}{\frac{4x}{x + 1}} = \frac{8x - 4}{4x} \]

### Step 6: Simplify the expression
Now, simplify the expression:
\[ f(f(x)) = \frac{8x - 4}{4x} = \frac{8x}{4x} - \frac{4}{4x} = 2 - \frac{1}{x} \]

### Final Answer
Thus, the final result is:
\[ f(f(x)) = 2 - \frac{1}{x} \]

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