Identifying Piecewise-Defined Functions from Graphs

Q1. Give a rule for the piecewise-defined function represented by the graph below.

Background

Topic: Piecewise-Defined Functions

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

Key Terms and Concepts:

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

• Open Circle: Indicates that the endpoint is not included (strict inequality, e.g., x < a or x > a).

• Closed Circle: Indicates that the endpoint is included (inclusive inequality, e.g., x ≤ a or x ≥ a).

Step-by-Step Guidance

1. Examine the graph and identify the different segments. Notice where the function changes its rule (these are usually at the x-values where open or closed circles appear).

2. For each segment, determine the equation of the line or curve. Use two points on each segment to find the slope and y-intercept if the segment is linear.

3. Write the domain for each segment using inequalities. Pay attention to open and closed circles to decide whether to use <, ≤, >, or ≥.

4. Combine your results into a piecewise function format: $f(x) = \begin{cases} \text{expression 1,} & \text{domain 1} \\ \text{expression 2,} & \text{domain 2} \\ \text{expression 3,} & \text{domain 3} \end{cases}$

Try solving on your own before revealing the answer!

Final Answer:

$f(x) = \begin{cases} 2x + 3, & x < -1 \\ 2, & -1 \leq x < 1 \\ x^2 + 1, & x \geq 1 \end{cases}$

The graph shows three distinct pieces: a line for $x < -1$, a constant for $-1 \leq x < 1$, and a parabola for $x \geq 1$. Open and closed circles indicate which intervals are included.