If `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 the function equal to \( y \)
We start by letting \( y = f(x) \):
\[ y = x^2 - x + 1 \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to express it in terms of \( x \):
\[ x^2 - x + (1 - y) = 0 \]
This is a quadratic equation in \( x \).

### Step 3: Identify coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we identify:
- \( a = 1 \)
- \( b = -1 \)
- \( c = 1 - y \)

### Step 4: Use the quadratic formula
We apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (1 - y)}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{1 \pm \sqrt{1 - 4(1 - y)}}{2} \]

### Step 5: Simplify the expression under the square root
Now, simplify the expression under the square root:
\[ x = \frac{1 \pm \sqrt{1 - 4 + 4y}}{2} \]
\[ x = \frac{1 \pm \sqrt{4y - 3}}{2} \]

### Step 6: Write the inverse function
Thus, we have two possible expressions for \( x \):
\[ x = \frac{1 + \sqrt{4y - 3}}{2} \quad \text{or} \quad x = \frac{1 - \sqrt{4y - 3}}{2} \]
We can express the inverse function as:
\[ f^{-1}(y) = \frac{1 \pm \sqrt{4y - 3}}{2} \]

### Step 7: Finalize the inverse function
Since \( f(x) \) is a quadratic function that opens upwards, it is not one-to-one over all real numbers. Therefore, we typically restrict the domain to ensure that the inverse is a function. We can choose:
\[ f^{-1}(y) = \frac{1 + \sqrt{4y - 3}}{2} \]
This is the principal branch of the inverse function.

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