Integrated Math 2: Piecewise Functions - Your Ultimate Guide

Piecewise Functions: Your Integrated Math 2 Guide 🚀

A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Let's break it down:

Definition and Notation 📚

A piecewise function is defined using different formulas for different intervals of its domain. The general form is:

$$f(x) =\begin{cases} f_1(x), & \text{if } x \in I_1 \\ f_2(x), & \text{if } x \in I_2 \\ ... & ... \\ f_n(x), & \text{if } x \in I_n \end{cases}$$

Where:

• $f_1(x), f_2(x), ..., f_n(x)$ are the sub-functions.
• $I_1, I_2, ..., I_n$ are the intervals in the domain.

Graphing Piecewise Functions 📈

To graph a piecewise function, graph each sub-function over its specified interval.

Example:

$$f(x) =\begin{cases} x + 2, & \text{if } x < 0 \\ x^2, & \text{if } 0 \leq x \leq 2 \\ 4, & \text{if } x > 2 \end{cases}$$

1. Graph $x + 2$ for $x < 0$: This is a line with a slope of 1 and a y-intercept of 2. Draw it only for $x < 0$. Use an open circle at $x = 0$.
2. Graph $x^2$ for $0 \leq x \leq 2$: This is a parabola. Draw it only between $x = 0$ and $x = 2$, including both endpoints (closed circles).
3. Graph $4$ for $x > 2$: This is a horizontal line at $y = 4$. Draw it only for $x > 2$. Use an open circle at $x = 2$.

Evaluating Piecewise Functions 🧐

To evaluate a piecewise function at a specific $x$ value, determine which interval $x$ belongs to, and then use the corresponding sub-function.

Example: Using the same function as above:

• $f(-1)$: Since $-1 < 0$, use $x + 2$. $f(-1) = -1 + 2 = 1$.
• $f(1)$: Since $0 \leq 1 \leq 2$, use $x^2$. $f(1) = 1^2 = 1$.
• $f(3)$: Since $3 > 2$, use $4$. $f(3) = 4$.

Code Example (Python) 💻

def piecewise_function(x):
    if x < 0:
        return x + 2
    elif 0 <= x <= 2:
        return x**2
    else:
        return 4

# Example usage
print(piecewise_function(-1))  # Output: 1
print(piecewise_function(1))   # Output: 1
print(piecewise_function(3))   # Output: 4

Real-World Applications 🌍

Piecewise functions are used in various real-world scenarios:

• Tax Brackets: Different income levels are taxed at different rates.
• Shipping Costs: Shipping costs may vary depending on the weight or size of the package.
• Utility Bills: Electricity or water rates may change based on usage.

Example: Tax Brackets 💰

Suppose the tax brackets are as follows:

• 10% for income up to $10,000
• 20% for income between $10,000 and $50,000
• 30% for income over $50,000

The tax owed can be modeled using a piecewise function:

$$Tax(income) =\begin{cases} 0.10 * income, & \text{if } income \leq 10000 \\ 0.20 * income, & \text{if } 10000 < income \leq 50000 \\ 0.30 * income, & \text{if } income > 50000 \end{cases}$$

This comprehensive guide should help you understand and work with piecewise functions in your Integrated Math 2 class. Good luck! 👍