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

Answer

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

### Step 1: Factor the quadratic expression
Start by factoring the quadratic expression inside the absolute value.

\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

### Step 2: Identify the roots
The roots of the equation \( x^2 - 3x + 2 = 0 \) are \( x = 1 \) and \( x = 2 \). This means that the quadratic expression changes sign at these points.

### Step 3: Analyze the sign of the quadratic expression
To determine where the expression \( x^2 - 3x + 2 \) is positive or negative, we can test intervals defined by the roots:

- For \( x < 1 \): Choose \( x = 0 \):
\[ 0^2 - 3(0) + 2 = 2 \quad (\text{positive}) \]

- For \( 1 < x < 2 \): Choose \( x = 1.5 \):
\[ (1.5)^2 - 3(1.5) + 2 = 2.25 - 4.5 + 2 = -0.25 \quad (\text{negative}) \]

- For \( x > 2 \): Choose \( x = 3 \):
\[ 3^2 - 3(3) + 2 = 9 - 9 + 2 = 2 \quad (\text{positive}) \]

### Step 4: Determine the behavior of \( f(x) \)
Now, we can summarize the sign of \( x^2 - 3x + 2 \):
- It is positive for \( x < 1 \) and \( x > 2 \).
- It is negative for \( 1 < x < 2 \).

Thus, we can express \( f(x) \) as:
\[ f(x) = \begin{cases} x^2 - 3x + 2 & \text{if } x < 1 \text{ or } x > 2 \\ -(x^2 - 3x + 2) & \text{if } 1 < x < 2 \end{cases} \]

### Step 5: Find the minimum value of \( f(x) \)
1. For \( x < 1 \) and \( x > 2 \), the minimum value occurs at the roots:
- At \( x = 1 \) and \( x = 2 \), \( f(1) = |0| = 0 \) and \( f(2) = |0| = 0 \).

2. For \( 1 < x < 2 \), since \( f(x) = -(x^2 - 3x + 2) \), we need to find the maximum of \( x^2 - 3x + 2 \) in this interval. The vertex of the parabola \( y = x^2 - 3x + 2 \) occurs at \( x = \frac{-b}{2a} = \frac{3}{2} \), which is in the interval \( (1, 2) \).

Calculating \( f(1.5) \):
\[ f(1.5) = |(1.5)^2 - 3(1.5) + 2| = |2.25 - 4.5 + 2| = |-0.25| = 0.25 \]

### Step 6: Determine the range
From our analysis:
- The minimum value of \( f(x) \) is \( 0 \) (at \( x = 1 \) and \( x = 2 \)).
- As \( x \) approaches \( -\infty \) or \( +\infty \), \( f(x) \) approaches \( +\infty \).

Thus, the range of \( f(x) \) is:
\[ \text{Range of } f(x) = [0, \infty) \]