Find `fof^(-1)` and `f^(-1)` of for the function :
`f(x)=(1)/(x),xne0`. Also prove that `fof^(-1)=f^(-1)of`.

Answer

To solve the problem, we need to find \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \) for the function \( f(x) = \frac{1}{x} \) where \( x \neq 0 \). We will also prove that \( f(f^{-1}(x)) = f^{-1}(f(x)) \).

### Step 1: Find the inverse function \( f^{-1}(x) \)

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

To find the inverse, we set \( y = f(x) \):
\[ y = \frac{1}{x} \]

Now, we solve for \( x \):
\[ x = \frac{1}{y} \]

Thus, the inverse function is:
\[ f^{-1}(x) = \frac{1}{x} \]

### Step 2: Find \( f(f^{-1}(x)) \)

Now we substitute \( f^{-1}(x) \) into \( f(x) \):
\[ f(f^{-1}(x)) = f\left(\frac{1}{x}\right) \]

Using the definition of \( f(x) \):
\[ f\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x}} = x \]

### Step 3: Find \( f^{-1}(f(x)) \)

Next, we substitute \( f(x) \) into \( f^{-1}(x) \):
\[ f^{-1}(f(x)) = f^{-1}\left(\frac{1}{x}\right) \]

Using the definition of \( f^{-1}(x) \):
\[ f^{-1}\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x}} = x \]

### Step 4: Prove that \( f(f^{-1}(x)) = f^{-1}(f(x)) \)

From the previous steps, we have:
\[ f(f^{-1}(x)) = x \]
\[ f^{-1}(f(x)) = x \]

Since both expressions equal \( x \), we can conclude that:
\[ f(f^{-1}(x)) = f^{-1}(f(x)) \]

### Final Result

Thus, we have found:
- \( f(f^{-1}(x)) = x \)
- \( f^{-1}(f(x)) = x \)
- Proved that \( f(f^{-1}(x)) = f^{-1}(f(x)) \)