Q1. Sketch the graph of the piecewise function:

\[ f(x) = \begin{cases} -\frac{1}{2}x^2 & \text{if } x < 1 \\ 2x + 1 & \text{if } x \geq 1 \end{cases} \]

Then, use the graph of to determine its range.

Background

Topic: Piecewise Functions and Range

This question tests your understanding of how to interpret and graph piecewise-defined functions, and how to use the graph to determine the range of the function.

Key Terms and Formulas

• Piecewise Function: A function defined by different expressions depending on the input value ().

• Range: The set of all possible output values () of a function.

Step-by-Step Guidance

1. Identify the two pieces of the function and their domains:

   • For , (a downward-opening parabola, left of ).

   • For , (a straight line, starting at and continuing right).

2. Sketch each piece on the coordinate plane:

   • For , plot the parabola up to but not including .

   • For , plot the line starting at (including $x=1$).

3. Check the value of each piece at the transition point :

   • Calculate for both pieces to see if the graph is continuous at or if there is a jump.

4. Use the graph to determine the lowest and highest -values for each piece, and look for any gaps in the range.

5. Combine the -values from both pieces to describe the overall range of .

Try solving on your own before revealing the answer!