Piecewise Functions from Graphs – Step-by-Step Guidance

Q5. Write a formula for the piecewise function graphed below.

Background

Topic: Piecewise Functions

This question tests your ability to interpret a graph and write a piecewise function that describes it. Piecewise functions are defined by different expressions depending on the input value (x).

Key Terms and Formulas

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

• Closed circle: Indicates the endpoint is included in the interval ("≤" or "≥").

• Open circle: Indicates the endpoint is not included ("<" or ">").

• Interval notation: Used to specify the domain for each piece.

Step-by-Step Guidance

1. Examine the graph and identify the intervals where the function changes. Notice the horizontal segments and their endpoints.

2. For each interval, determine the constant value of the function. For example, the segment from $x = -5$ to $x = -1$ is at $y = 2$.

3. Pay attention to whether the endpoints are included (closed circle) or excluded (open circle) for each interval. This will affect the inequality signs in your domain restrictions.

4. Write the function as a piecewise formula, using the correct values and domain restrictions for each segment. For example, $f(x) = 2$ for $-5 \leq x \leq -1$.

5. Repeat this process for the other segments: $y = -2$ from $x = -1$ to $x = 2$, and $y = -4$ from $x = 2$ to $x = 4$.

Try solving on your own before revealing the answer!

Final Answer:

$f(x) = \begin{cases} 2 & \text{if } -5 \leq x \leq -1 \\ -2 & \text{if } -1 < x \leq 2 \\ -4 & \text{if } 2 < x \leq 4 \end{cases}$

This formula matches the graph: each segment is constant, and the domain restrictions reflect the open/closed circles at the endpoints.