Analyzing a Piecewise Function and Its Range

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 $f$ 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 ($x$).

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

Step-by-Step Guidance

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

   • For $x < 1$, $f(x) = -\frac{1}{2}x^2$ (a downward-opening parabola, left of $x=1$).

   • For $x \geq 1$, $f(x) = 2x + 1$ (a straight line, starting at $x=1$ and continuing right).

2. Sketch each piece on the coordinate plane:

   • For $x < 1$, plot the parabola up to but not including $x=1$.

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

3. Check the value of each piece at the transition point $x=1$:

   • Calculate $f(1)$ for both pieces to see if the graph is continuous at $x=1$ or if there is a jump.

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

5. Combine the $y$-values from both pieces to describe the overall range of $f(x)$.

Try solving on your own before revealing the answer!

Final Answer:

The range of $f(x)$ is $(-\infty, 0] \cup [3, \infty)$.

The parabola covers all $y$-values up to $0$(not including values above $0$), and the line starts at $y=3$ and increases without bound.