If `f(x-1)=f(x+1)`, where `f(x)=x^2-2x+3`, then: `x=`

Options

A. 1
B. 2
C. 3
D. none of these.

Answer

To solve the equation \( f(x-1) = f(x+1) \) where \( f(x) = x^2 - 2x + 3 \), we will follow these steps:

### Step 1: Write down the function
We have the function:
\[ f(x) = x^2 - 2x + 3 \]

### Step 2: Substitute \( x-1 \) into the function
Now, we need to find \( f(x-1) \):
\[ f(x-1) = (x-1)^2 - 2(x-1) + 3 \]
Expanding this:
\[ = (x^2 - 2x + 1) - (2x - 2) + 3 \]
\[ = x^2 - 2x + 1 - 2x + 2 + 3 \]
\[ = x^2 - 4x + 6 \]

### Step 3: Substitute \( x+1 \) into the function
Next, we find \( f(x+1) \):
\[ f(x+1) = (x+1)^2 - 2(x+1) + 3 \]
Expanding this:
\[ = (x^2 + 2x + 1) - (2x + 2) + 3 \]
\[ = x^2 + 2x + 1 - 2x - 2 + 3 \]
\[ = x^2 + 2 \]

### Step 4: Set the two expressions equal to each other
Now we set \( f(x-1) \) equal to \( f(x+1) \):
\[ x^2 - 4x + 6 = x^2 + 2 \]

### Step 5: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ -4x + 6 = 2 \]
Now, isolate \( x \):
\[ -4x = 2 - 6 \]
\[ -4x = -4 \]
\[ x = 1 \]

### Conclusion
The value of \( x \) is:
\[ \boxed{1} \]