Special Functions

1️⃣ Piecewise Function

What is a Piecewise Function?

A piecewise function has different definitions depending on the interval. Its definition might be different when x > 0 compared to when x < 0, or it might have specific definitions for different ranges like 1 to 5 versus 5 to 10.

Example: A Piecewise Function

f(x) = {
x + 2, if x < 0
x, if x ≥ 0

1When x < 0 (negative side): Use the equation f(x) = x + 2

2When x ≥ 0 (positive side and zero): Use the equation f(x) = x

The left side follows x + 2, the right side follows x
Notice the jump at x = 0

2️⃣ Greatest Integer Function (Floor Function)

Understanding the Greatest Integer Function

To understand this function, imagine a strict teacher who said: "I won't accept any fractions. Any student who gets a grade will have the fraction removed."

📚 A student who gets 1.5 → gets 1
📚 A student who gets 1.25 → gets 1
📚 A student who gets 2.0 → stays at 2

Since we're being strict, instead of rounding up, we round down to the lower number.

How the Greatest Integer Function Works

f(x) = ⌊x⌋

1For all numbers from 0 to 1: f(x) = 0

2For all numbers from 1 to 2: f(x) = 1

3For all numbers from 2 to 3: f(x) = 2

4And so on...

The function creates step-like horizontal segments
Each step represents a constant integer value

3️⃣ Absolute Value Function

What is the Absolute Value Function?

The absolute value function is written as f(x) = |x|. Its job is to eliminate negative signs and always give a non-negative result.

How Absolute Value Works

f(x) = |x|

1On the positive side (x ≥ 0): f(x) = x (nothing changes)

2On the negative side (x < 0): The absolute value removes the negative sign

3Result: We get a V-shaped graph that mirrors the positive side

The graph forms a V-shape
The negative side is a mirror reflection of the positive side

Key Points Summary

✓Piecewise functions have different rules for different intervals of x
✓Greatest integer function rounds down to the nearest integer (floor function)
✓Absolute value function always returns non-negative values
✓Each function type creates a distinctive graph shape
✓These functions are used in many real-world applications
✓Understanding the domain and behavior is key to graphing these functions

FLOOR FUNCTION-PIECEWISE FUNCTIONS & ABSOLUTE VALUE FUNCTION