If `f (x) = x^(2) -2x + 3,` then the value of x for which `f (x) = f(x +1)` is

Options

A. `1//2`
B. `1//3`
C. `1`
D. `3`

Answer

To solve the problem, we need to find the value of \( x \) for which \( f(x) = f(x + 1) \) given the function \( f(x) = x^2 - 2x + 3 \).

### Step-by-Step Solution:

1. **Write down the function**:
\[ f(x) = x^2 - 2x + 3 \]

2. **Calculate \( f(x + 1) \)**:
To find \( f(x + 1) \), we substitute \( x + 1 \) into the function:
\[ f(x + 1) = (x + 1)^2 - 2(x + 1) + 3 \]
Expanding this:
\[ = (x^2 + 2x + 1) - (2x + 2) + 3 \]
Simplifying further:
\[ = x^2 + 2x + 1 - 2x - 2 + 3 \]
\[ = x^2 + 1 + 1 = x^2 + 2 \]

3. **Set the two functions equal**:
Now we set \( f(x) \) equal to \( f(x + 1) \):
\[ x^2 - 2x + 3 = x^2 + 2 \]

4. **Simplify the equation**:
Subtract \( x^2 \) from both sides:
\[ -2x + 3 = 2 \]
Now, isolate \( x \):
\[ -2x = 2 - 3 \]
\[ -2x = -1 \]
Dividing both sides by -2:
\[ x = \frac{-1}{-2} = \frac{1}{2} \]

5. **Final answer**:
The value of \( x \) for which \( f(x) = f(x + 1) \) is:
\[ x = \frac{1}{2} \]