Let `f:(-oo, 1]rarr(-oo, 1]` such that `f(x)=x(2-x)`, then `f^(-1)(x)` is :

Options

A. a. `1+sqrt(1-x)`
B. b. `1-sqrt(1-x)`
C. c. `sqrt(1-x)`
D. d. none of these

Answer

To find the inverse function \( f^{-1}(x) \) for the function \( f(x) = x(2 - x) \), we will follow these steps:

### Step 1: Set up the equation
We start with the function:
\[ y = f(x) = x(2 - x) \]
This can be rewritten as:
\[ y = 2x - x^2 \]

### Step 2: Rearrange the equation
Rearranging gives us:
\[ x^2 - 2x + y = 0 \]
This is a quadratic equation in terms of \( x \).

### Step 3: Apply the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -2, c = y \):
\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot y}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{2 \pm \sqrt{4 - 4y}}{2} \]
\[ x = \frac{2 \pm 2\sqrt{1 - y}}{2} \]
\[ x = 1 \pm \sqrt{1 - y} \]

### Step 4: Determine the correct sign
Since the function \( f(x) \) is defined from \( (-\infty, 1] \), we need to take the negative root to ensure that we stay within the range of the function:
\[ x = 1 - \sqrt{1 - y} \]

### Step 5: Solve for \( f^{-1}(x) \)
Now we can express the inverse function:
\[ f^{-1}(x) = 1 - \sqrt{1 - x} \]

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

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