Piecewise Functions: Writing Equations from Graphs

Q1. Write the equation of the piecewise-defined function f(x) whose graph is given 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 Formulas

• 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 in the interval (use < or >).

• Closed Circle: Indicates that the endpoint is included in the interval (use ≤ or ≥).

• General Form:

$f(x) = \begin{cases} \\ \text{expression 1,} & \text{if } x \text{ in interval 1} \\ \text{expression 2,} & \text{if } x \text{ in interval 2} \\ \text{expression 3,} & \text{if } x \text{ in interval 3} \\ \end{cases}$

Step-by-Step Guidance

1. Examine each segment of the graph and identify the intervals for x. Pay close attention to open and closed circles to determine whether endpoints are included.

2. For each interval, determine the equation of the segment. For example, if the segment is a horizontal line, the equation will be of the form $y = c$; if it is a parabola, it may be $y = ax^2 + bx + c$.

3. Write the function for each interval, using the correct domain restrictions (e.g., $x < a$, $a \leq x < b$, $x \geq b$).

4. Combine the expressions into a single piecewise function, making sure each interval matches the graph's behavior and endpoints.

Try solving on your own before revealing the answer!

Final Answer:

$f(x) = \begin{cases} \\ 7 - x, & x < -2 \\ x^2 - 5, & -2 \leq x < 2 \\ 3, & x \geq 2 \\ \end{cases}$

The function is defined by three pieces: a linear segment for $x < -2$, a quadratic for $-2 \leq x < 2$, and a constant for $x \geq 2$. The open and closed circles on the graph indicate which intervals include their endpoints.