Which of the following functions is inverse of itself?

Options

A. (A)`f(x)=(1-x)/(1+x)`
B. (B)`g(x)=5^(log x)`
C. (C)`h(x)=2^(x(x-1))`
D. (D)None of these

Answer

To determine which of the given functions is its own inverse, we need to analyze each function and check if \( f(f(x)) = x \). Let's go through the options step by step.

### Step 1: Analyze Option A
The function given is:
\[ f(x) = \frac{1 - x}{1 + x} \]
To find the inverse, we set \( y = f(x) \):
\[ y = \frac{1 - x}{1 + x} \]
Now, we need to solve for \( x \) in terms of \( y \).

### Step 2: Solve for \( x \) in terms of \( y \)
Multiply both sides by \( 1 + x \):
\[ y(1 + x) = 1 - x \]
Expanding this gives:
\[ y + yx = 1 - x \]
Rearranging the equation:
\[ yx + x = 1 - y \]
Factoring out \( x \):
\[ x(y + 1) = 1 - y \]
Thus, we find:
\[ x = \frac{1 - y}{1 + y} \]

### Step 3: Substitute \( y \) back to find \( f^{-1}(x) \)
Now, replace \( y \) with \( x \):
\[ f^{-1}(x) = \frac{1 - x}{1 + x} \]

### Step 4: Compare \( f(x) \) and \( f^{-1}(x) \)
We see that:
\[ f^{-1}(x) = f(x) \]
This means that \( f(x) \) is indeed its own inverse.

### Step 5: Analyze Option B
The function given is:
\[ g(x) = 5^{\log(x)} \]
Using properties of logarithms, we know:
\[ g(x) = x \]
This function is not its own inverse.

### Step 6: Analyze Option C
The function given is:
\[ h(x) = 2^x(x - 1) \]
To find the inverse, we set \( y = h(x) \):
\[ y = 2^x(x - 1) \]
This is a more complex function, and we would find that it does not simplify to \( h(h(x)) = x \).

### Conclusion
After analyzing all options, we find that the only function that is its own inverse is:
\[ \text{Option A: } f(x) = \frac{1 - x}{1 + x} \]