Graph each piecewise-defined function.

Verified step by step guidance

1. Identify the three pieces of the piecewise function and their corresponding domains: \(f(x) = \begin{cases} 3x + 7, & \text{if } x < -4 \\ -2x + 2, & \text{if } -4 \leq x \leq 3 \\ 5x - 10, & \text{if } x > 3 \end{cases}\)
2. For the first piece, \(f(x) = 3x + 7\) when \(x < -4\), plot the line \(y = 3x + 7\) only for values of \(x\) less than \(-4\). Use an open circle at \(x = -4\) to indicate that this point is not included in this piece.
3. For the second piece, \(f(x) = -2x + 2\) when \(-4 \leq x \leq 3\), plot the line \(y = -2x + 2\) for \(x\) values from \(-4\) to \(3\) inclusive. Use closed circles at \(x = -4\) and \(x = 3\) to show these points are included.
4. For the third piece, \(f(x) = 5x - 10\) when \(x > 3\), plot the line \(y = 5x - 10\) for \(x\) values greater than \(3\). Use an open circle at \(x = 3\) to indicate this point is not included in this piece.
5. Combine all three pieces on the same coordinate plane, making sure to respect the domain restrictions and endpoint types (open or closed circles) to accurately represent the piecewise function.

Key Concepts

Piecewise-Defined Functions

A piecewise-defined function is a function composed of multiple sub-functions, each applying to a specific interval of the domain. Understanding how to interpret and graph each piece separately is essential for analyzing the overall function.

Graphing Linear Functions

Each piece of the piecewise function is a linear function of the form y = mx + b. Knowing how to graph linear functions by identifying slope and y-intercept helps in plotting each segment accurately on the coordinate plane.

Domain Restrictions and Continuity

Each sub-function applies only within a specified domain interval. Recognizing these domain restrictions ensures correct graphing boundaries and helps determine if the function is continuous or has breaks at the interval endpoints.