If `f(x)` is a one to one function, where `f(x)=x^(2)-x+1`, then find the inverse of the `f(x):`

Options

A. `(x-(1)/(2))`
B. `sqrt(x-(3)/(4))+(1)/(2)`
C. `sqrt(x-(3)/(4))-(1)/(2)`
D. none of these

Answer

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

### Step 1: Set \( f(x) \) equal to \( y \)
We start by letting \( y = f(x) \):
\[ y = x^2 - x + 1 \]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, we swap \( x \) and \( y \):
\[ x = y^2 - y + 1 \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ y^2 - y + (1 - x) = 0 \]
This is a quadratic equation in terms of \( y \).

### Step 4: Use the quadratic formula
We can use the quadratic formula to solve for \( y \):
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \( a = 1 \), \( b = -1 \), and \( c = 1 - x \).

### Step 5: Substitute values into the quadratic formula
Substituting the values into the formula, we have:
\[ y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (1 - x)}}{2 \cdot 1} \]
This simplifies to:
\[ y = \frac{1 \pm \sqrt{1 - 4(1 - x)}}{2} \]

### Step 6: Simplify the expression under the square root
Simplifying the expression under the square root:
\[ y = \frac{1 \pm \sqrt{1 - 4 + 4x}}{2} \]
This further simplifies to:
\[ y = \frac{1 \pm \sqrt{4x - 3}}{2} \]

### Step 7: Determine the inverse function
Thus, we have two possible values for \( y \):
\[ y = \frac{1 + \sqrt{4x - 3}}{2} \quad \text{and} \quad y = \frac{1 - \sqrt{4x - 3}}{2} \]
Since \( f(x) \) is a one-to-one function, we will choose the positive root:
\[ f^{-1}(x) = \frac{1 + \sqrt{4x - 3}}{2} \]

### Final Answer
The inverse of the function \( f(x) = x^2 - x + 1 \) is:
\[ f^{-1}(x) = \frac{1 + \sqrt{4x - 3}}{2} \]

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