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Piecewise functions are mathematical expressions that define a function using different formulas over different intervals. Imagine a road that has speed limits that vary depending on where you are along the path. Similarly, a piecewise function tells you which mathematical rule applies depending on the value of the input variable.

• It consists of intervals expressed as conditions.
• Each interval has a specific expression or formula.

In our exercise, we have a function that behaves differently depending on whether the variable \( b \) is greater than or less than 3. By using piecewise notation, you can clearly see how the function responds to different inputs, making it easier to understand discontinuous or multi-rule functions.

When dealing with absolute values, it’s key to understand positive and negative scenarios. Absolute values measure the distance a number is from zero, always giving a non-negative result. To rewrite an absolute value expression, we consider both positive and negative cases of the expression inside the absolute value.
- **Positive Scenario:** If \( b \geq 3 \), then \( |b - 3| = b - 3 \). Here, the expression inside the absolute value is already non-negative, so we can safely drop the absolute value symbols.
- **Negative Scenario:** If \( b < 3 \), the expression \( b - 3 \) is negative. To find its absolute value, we multiply by -1: \( |b - 3| = -(b - 3) \). This operation flips the sign, making it positive without changing its magnitude.

Simplifying expressions without absolute values involves translating each condition into a basic algebraic expression. By doing this, you remove the absolute value bars and represent the number’s absolute distance under varying conditions.

• **For \( b \geq 3 \):** The expression becomes \( b - 3 \).
• **For \( b < 3 \):** The expression transforms into \( -(b - 3) \), or equivalently \( 3 - b \).

This approach aids in graphing and solving equations, as you work with simple, linear expressions instead of dealing with complex absolute value symbols.

Solving problems with absolute values using a step-by-step approach helps ensure thorough understanding and accuracy.

Step 1: Identify Conditions

Start by identifying when the expression inside the absolute value is positive or negative. Here, the conditions are \( b \geq 3 \) and \( b < 3 \).

Step 2: Transform the Expression

Convert the absolute value expression based on these conditions:

• For \( b \geq 3 \), simply take \( b - 3 \).
• For \( b < 3 \), use \( -(b - 3) \).

Step 3: Combine and Represent

Finally, compile these transformations into a piecewise function:\[\begin{cases} b-3 & \text{if } b \geq 3 \ 3-b & \text{if } b < 3\end{cases}\]Using these steps enables a clear breakdown of how absolute values can be resolved into separate linear components.