If `f (x) =1 -1/x, ` then `f (f ((1)/(x)))` is

Options

A. `1/x`
B. `(1)/(1+x)`
C. `(x)/(x-1)`
D. `(1)/(x-1)`

Answer

To solve the problem step by step, we need to find \( f(f(1/x)) \) given that \( f(x) = 1 - \frac{1}{x} \).

### Step 1: Find \( f(1/x) \)

We start by substituting \( x \) with \( \frac{1}{x} \) in the function \( f(x) \).

\[ f\left(\frac{1}{x}\right) = 1 - \frac{1}{\frac{1}{x}} = 1 - x \]

### Step 2: Find \( f(f(1/x)) \)

Now we need to find \( f(f(1/x)) \), which means we need to substitute \( f(1/x) \) into the function \( f \).

From Step 1, we have \( f(1/x) = 1 - x \). Now we substitute this into \( f \):

\[ f(f(1/x)) = f(1 - x) \]

### Step 3: Substitute \( 1 - x \) into \( f(x) \)

Now we substitute \( 1 - x \) into the function \( f(x) \):

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

### Step 4: Simplify \( f(1 - x) \)

Next, we simplify \( 1 - \frac{1}{1 - x} \):

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

### Step 5: Final Result

Thus, we have:

\[ f(f(1/x)) = \frac{-x}{1 - x} \]

### Final Answer

The final answer is:

\[ f(f(1/x)) = \frac{-x}{1 - x} \]

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