The value of the parameter a for which the function f (x) = 1 + ` alpha x, alpha ne 0 ` is the inverse of itself, is

Options

A. `-2`
B. `-1`
C. `1`
D. `2`

Answer

To find the value of the parameter \( \alpha \) for which the function \( f(x) = 1 + \alpha x \) is its own inverse, we will follow these steps:

### Step 1: Set up the function
We start with the function:
\[ f(x) = 1 + \alpha x \]

### Step 2: Find the inverse function
To find the inverse function, we first set \( f(x) \) equal to \( y \):
\[ y = 1 + \alpha x \]
Next, we solve for \( x \) in terms of \( y \):
\[ \alpha x = y - 1 \]
\[ x = \frac{y - 1}{\alpha} \]
Now, we replace \( y \) with \( f^{-1}(x) \):
\[ f^{-1}(x) = \frac{x - 1}{\alpha} \]

### Step 3: Set the function equal to its inverse
Since we want the function to be its own inverse, we set \( f(x) \) equal to \( f^{-1}(x) \):
\[ 1 + \alpha x = \frac{x - 1}{\alpha} \]

### Step 4: Clear the fraction
To eliminate the fraction, we multiply both sides by \( \alpha \):
\[ \alpha (1 + \alpha x) = x - 1 \]
Expanding the left side gives:
\[ \alpha + \alpha^2 x = x - 1 \]

### Step 5: Rearrange the equation
Rearranging the equation to one side, we have:
\[ \alpha^2 x - x + \alpha + 1 = 0 \]
Factoring out \( x \):
\[ (\alpha^2 - 1)x + (\alpha + 1) = 0 \]

### Step 6: Solve for \( x \)
For the equation to hold for all \( x \), both coefficients must equal zero:
1. \( \alpha^2 - 1 = 0 \)
2. \( \alpha + 1 = 0 \)

### Step 7: Solve the equations
From \( \alpha^2 - 1 = 0 \):
\[ \alpha^2 = 1 \implies \alpha = 1 \text{ or } \alpha = -1 \]

From \( \alpha + 1 = 0 \):
\[ \alpha = -1 \]

### Step 8: Determine the valid solution
Since \( \alpha \neq 0 \) is given, both \( \alpha = 1 \) and \( \alpha = -1 \) are valid. However, we need to check which of these makes the function its own inverse.

1. For \( \alpha = 1 \):
\[ f(x) = 1 + x \quad \text{and} \quad f^{-1}(x) = x - 1 \quad \text{(not equal)} \]

2. For \( \alpha = -1 \):
\[ f(x) = 1 - x \quad \text{and} \quad f^{-1}(x) = 1 - x \quad \text{(equal)} \]

Thus, the value of \( \alpha \) for which the function is its own inverse is:
\[ \alpha = -1 \]

### Final Answer:
\[ \alpha = -1 \]