A piecewise function is a function that is defined by multiple sub-functions, each of which applies to a specific interval or domain. Instead of having a single expression for the entire domain, the output of the function depends on which part of the domain (or interval) the input value belongs to.

Definition

For example, a piecewise function $f(x)$ can be written as:

\begin{cases} f_1(x), & \text{if } x \in \text{Interval 1} \\ f_2(x), & \text{if } x \in \text{Interval 2} \\ \vdots \\ f_n(x), & \text{if } x \in \text{Interval n} \end{cases}$$ Each function $$f_1(x), f_2(x), \ldots, f_n(x)$$ governs the output for different intervals of $$x$$. ### Example Consider the following piecewise function: $$f(x) = \begin{cases} 2x + 1, & \text{if } x < 0 \\ x^2, & \text{if } 0 \leq x \leq 2 \\ 5, & \text{if } x > 2 \end{cases}$$ This means: - When $$x$$ is less than 0, the output is determined by the function $$2x + 1$$. - When $$x$$ is between 0 and 2 (inclusive), the output is determined by $$x^2$$. - When $$x$$ is greater than 2, the output is always 5. ### Why Use Piecewise Functions? Piecewise functions are useful for modeling situations where a process or rule changes based on conditions. For example: - Tax brackets in a progressive tax system can be modeled with a piecewise function. - Speed limits in different zones can also be modeled as piecewise functions. Would you like further details or have any specific questions? ### Related Questions: 1. How do you find the domain and range of a piecewise function? 2. Can a piecewise function be continuous? What does that mean? 3. How do you graph a piecewise function? 4. How do you compute limits for piecewise functions at boundary points? 5. Can you explain the derivative of a piecewise function? ### Tip: When graphing a piecewise function, it helps to evaluate each piece separately and check if the function has any jumps or discontinuities.