A piecewise function is a function in which formula is used to define the output

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:

\[
f(x) = 
\begin{cases} 
f_1(x), & \text{if } x \in \text{Interval 1} \\
f_2(x), & \text{if } x \in \text{Interval 2} \\
\vdots \\
f_n(x), & \text{if } x \in \text{Interval n}
\end{cases}
\]

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.

A piecewise function is a function in which formula is used to define the output.

Explore the concept of piecewise functions, which are defined by multiple sub-functions over different intervals. Learn how to interpret and solve problems involving piecewise functions with detailed examples, including their application in algebra and real-world scenarios. Suitable for Grades 9-12.

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