math

Definition: A piecewise function is a function that is defined by multiple sub-functions, each of which applies to a certain interval of the independent variable.
General Form:
f(x) = \begin{cases}
f1(x) & \text{if } x < a \ f2(x) & \text{if } a \leq x < b \
f_3(x) & \text{if } x \geq b
\end{cases}
Example:
f(x) = \begin{cases}
x^2 & \text{if } x < 0 \
2x + 1 & \text{if } 0 \leq x < 3 \
5 & \text{if } x \geq 3
\end{cases}
Graphing:
- Graph each piece separately and ensure to use open or closed circles to denote whether the endpoints are included in each interval.

Definition: A linear function is a function that can be expressed in the form of $f(x) = mx + b$ , where $m$ represents the slope and $b$ the y-intercept.
Slope (m): The slope of a line indicates its steepness and direction. It is calculated as:
$m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
Y-Intercept (b): The y-intercept is where the line crosses the y-axis (i.e., when $x = 0$ ).
Graphing:
- Start at the y-intercept (b) on the y-axis.
- From there, use the slope (m) to find other points on the line. For example, if the slope is $2$ , rise 2 units up and run 1 unit to the right to find another point.

Form: Piecewise functions can contain multiple definitions based on the value of x, whereas linear functions maintain a constant slope throughout their domain.
Graph Shape: Piecewise functions can appear broken or segmented due to their multiple definitions, while linear functions produce straight lines.

Continuity: Check for continuity at the boundaries in piecewise functions.
Domain and Range: Identify the domain and range for both piecewise and linear functions.