Find the range of
`f(x)=x^(2)-2|x|+3`

Answer

To find the range of the function \( f(x) = x^2 - 2|x| + 3 \), we can follow these steps:

### Step 1: Rewrite the function using a substitution
Since the function involves the absolute value \( |x| \), we can simplify our analysis by substituting \( t = |x| \). Thus, we can rewrite the function as:
\[ f(t) = t^2 - 2t + 3 \]
where \( t \geq 0 \).

### Step 2: Analyze the quadratic function
The function \( f(t) = t^2 - 2t + 3 \) is a quadratic function. To find its range, we need to determine its vertex, since a quadratic function opens upwards (the coefficient of \( t^2 \) is positive).

### Step 3: Find the vertex of the quadratic
The vertex of a quadratic function in the form \( at^2 + bt + c \) can be found using the formula:
\[ t = -\frac{b}{2a} \]
For our function, \( a = 1 \) and \( b = -2 \):
\[ t = -\frac{-2}{2 \cdot 1} = 1 \]

### Step 4: Calculate the value of the function at the vertex
Now we substitute \( t = 1 \) back into the function to find the minimum value:
\[ f(1) = 1^2 - 2 \cdot 1 + 3 = 1 - 2 + 3 = 2 \]

### Step 5: Determine the range
Since the quadratic opens upwards and the minimum value occurs at \( t = 1 \) with \( f(1) = 2 \), the function will take all values from this minimum to infinity. Therefore, the range of \( f(t) \) is:
\[ [2, \infty) \]

### Step 6: Relate back to the original function
Since \( t = |x| \) can take any non-negative value, the range of the original function \( f(x) = x^2 - 2|x| + 3 \) remains the same:
\[ \text{Range of } f(x) = [2, \infty) \]

### Final Answer
The range of the function \( f(x) = x^2 - 2|x| + 3 \) is:
\[ [2, \infty) \]