Let h(x) = |x2 − 4x + 3|. Rewrite h(x) as a piecewise function using the definition of absolute value. Simplify the position of each subfunction.

To rewrite the function \( h(x) = |x^2 - 4x + 3| \) as a piecewise function, we need to follow the definition of absolute value. The absolute value of a function \( f(x) \) is defined as:

\[
|f(x)| = 
\begin{cases}
f(x), & \text{if } f(x) \geq 0, \\
-f(x), & \text{if } f(x) < 0.
\end{cases}
\]

### Step 1: Factorize the quadratic expression inside the absolute value
First, rewrite the quadratic expression:

\[
x^2 - 4x + 3 = (x - 1)(x - 3).
\]

### Step 2: Find the critical points
The critical points of this quadratic function occur when \( x^2 - 4x + 3 = 0 \), i.e., at \( x = 1 \) and \( x = 3 \). These points divide the real number line into three intervals: \( (-\infty, 1] \), \( [1, 3] \), and \( [3, \infty) \).

### Step 3: Determine the sign of \( x^2 - 4x + 3 \) on each interval
To determine the sign of the quadratic expression in each interval:

1. **For \( x \in (-\infty, 1] \):**  
   \( (x - 1)(x - 3) \geq 0 \) is **positive** (both factors are negative), so the expression \( x^2 - 4x + 3 \geq 0 \).
   
2. **For \( x \in [1, 3] \):**  
   \( (x - 1)(x - 3) \leq 0 \) is **negative** (one factor is positive and the other is negative), so the expression \( x^2 - 4x + 3 \leq 0 \).

3. **For \( x \in [3, \infty) \):**  
   \( (x - 1)(x - 3) \geq 0 \) is **positive** (both factors are positive), so the expression \( x^2 - 4x + 3 \geq 0 \).

### Step 4: Write the piecewise function
Now, we apply the absolute value definition:

\[
h(x) = 
\begin{cases}
x^2 - 4x + 3, & \text{if } x \leq 1 \text{ or } x \geq 3, \\
-(x^2 - 4x + 3), & \text{if } 1 < x < 3.
\end{cases}
\]

### Step 5: Simplify the expression
Simplify the negative part:

\[
-(x^2 - 4x + 3) = -x^2 + 4x - 3.
\]

Thus, the piecewise function is:

\[
h(x) = 
\begin{cases}
(x - 1)(x - 3), & \text{if } x \leq 1 \text{ or } x \geq 3, \\
-(x - 1)(x - 3), & \text{if } 1 < x < 3.
\end{cases}
\]

This is the simplified piecewise form of the function.

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Would you like further details on how the intervals were determined or how to analyze these types of functions?

Here are 5 related questions:
1. How can we graph the piecewise function \( h(x) \)?
2. What are the critical points and why are they important?
3. How do you determine the sign of a quadratic expression?
4. Can you describe the behavior of quadratic functions with respect to absolute value?
5. What is the difference between solving inequalities and equalities in piecewise functions?

**Tip:** When solving piecewise functions involving absolute values, always identify the critical points where the expression inside the absolute value changes sign.

Let h(x) = |x^2 − 4x + 3|. Rewrite h(x) as a piecewise function using the definition of absolute value. Simplify the position of each subfunction.

Learn how to rewrite the absolute value function h(x) = |x^2 - 4x + 3| as a piecewise function. This solution covers key algebraic concepts, including factoring quadratics and applying the definition of absolute value. Suitable for students in Grades 9-12.

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