Q1. The graph of a piecewise-defined function is given. Write a definition for the function.

Background

Topic: Piecewise-Defined Functions

This question is testing your ability to interpret a graph and write a piecewise-defined function that matches the graph. Piecewise functions are defined by different expressions depending on the interval of the input variable (x).

Key Terms and Formulas

• Piecewise Function: A function defined by multiple sub-functions, each applying to a certain interval of the domain.

• Interval Notation: Used to specify the domain for each piece of the function.

• Linear Function: A function of the form .

Step-by-Step Guidance

1. Observe the graph and identify the intervals where the function changes its rule. Notice the points where the graph changes direction or has endpoints.

2. Determine the equation of the line segment for each interval. Use the coordinates of the endpoints to find the slope () and y-intercept () for each segment using the formula:

   Then use to find the equation for each segment.

3. Write the piecewise function using the equations found in step 2, and specify the domain for each piece based on the x-values of the endpoints.

4. Check that the function values at the endpoints match the graph (including whether the endpoints are open or closed circles).

Try solving on your own before revealing the answer!