Piecewise Functions: Writing Equations and Graphing

Q1. Write the equation of the piecewise function shown in each graph.

Background

Topic: Piecewise Functions

This question tests your ability to interpret and write equations for piecewise functions based on their graphs. Piecewise functions are defined by different expressions depending on the value of the input variable (usually x).

Key Terms and Formulas:

• Piecewise Function: A function composed of two or more "pieces" (expressions), each applying to a specific interval of the domain.

• Interval Notation: Specifies the range of x-values for which each piece applies.

• Function Notation: $f(x) = \begin{cases} \text{expression}_1, & \text{condition}_1 \\ \text{expression}_2, & \text{condition}_2 \end{cases}$

Step-by-Step Guidance

1. Examine the graph and identify the distinct segments. Each segment corresponds to a different piece of the function.

2. For each segment, determine the equation (linear or otherwise) that describes the segment. Look for slope and y-intercept if the segment is linear.

3. Identify the domain (interval of x-values) for each segment. This is where the piece applies.

4. Write the piecewise function using the equations and intervals you found. Use proper notation to show which equation applies to which interval.

5. Check for open or closed circles on the graph to determine whether the endpoints are included or excluded in each interval.

Try solving on your own before revealing the answer!

Final Answer:

For the first graph (image_2):

$f(x) = \begin{cases} x - 3, & x \leq -4 \\ x + 3, & x > -1 \end{cases}$

For the second graph (image_3):

$f(x) = \begin{cases} 1, & x < -3 \\ -x - 1, & -3 \leq x \leq 3 \\ -6, & x > 3 \end{cases}$

Each piece matches the segment and interval shown in the graph. The open and closed circles indicate whether endpoints are included.