Function Fundamentals

🎯 Functions: Your Magic Input-Output Machine!

The Story Begins…

Imagine you have a magical vending machine. You put in a coin (input), press a button, and out pops exactly one snack (output). Every time you put in the same coin and press the same button, you get the exact same snack. No surprises!

That’s a function! A reliable machine that takes something in and gives something specific out.

🤖 What is a Function?

A function is like a loyal robot helper. You give it something, it follows its one rule, and gives you back exactly one answer.

The Three Magic Rules:

1. Every input gets an output — the robot never ignores you
2. Same input = same output — the robot is consistent
3. One input = ONE output only — the robot never gives two answers at once

Think of it like this:

INPUT → 🤖 FUNCTION → OUTPUT

   2   →  [Double it]  →   4
   5   →  [Double it]  →  10

Real Life Example:

• Your age is a function of your birth year
• Put in 2015 → Get back 10 (in 2025)
• Same birth year always gives same age!

✏️ Function Notation: The Robot’s Name Tag

Instead of saying “the doubling robot,” mathematicians write:

f(x) = 2x

Let’s break this down:

• f = the function’s name (like naming a pet!)
• (x) = the input slot
• = 2x = the rule (multiply by 2)

🎨 Different Names, Different Jobs

f(x) = x + 5      ← adds 5 to any number
g(x) = x²         ← squares any number
h(t) = 3t - 1     ← uses 't' instead of 'x'

Pro tip: The letter doesn’t matter! f(x), f(t), f(🍕) all work. It’s just a placeholder for “whatever you put in.”

🧮 Evaluating Functions: Feeding the Machine

Evaluating means: “I’m putting THIS number in. What comes out?”

Example: If f(x) = 3x + 2

Question: What is f(4)?

Solution:

f(x) = 3x + 2
f(4) = 3(4) + 2    ← Replace x with 4
f(4) = 12 + 2
f(4) = 14          ✓

Another Example: If g(x) = x² - 1

Question: What is g(5)?

g(5) = 5² - 1
g(5) = 25 - 1
g(5) = 24          ✓

You’re basically asking: “Hey robot, what do you give me when I give you 5?”

🚪 Domain and Range: Entry Rules & Exit Results

Domain = Who Can Enter? 🎫

The domain is all the numbers you’re ALLOWED to put into the function.

Think of a movie theater:

• Kids under 5 = FREE (allowed)
• Everyone else = needs a ticket
• Pets = NOT ALLOWED ❌

Range = What Comes Out? 🎁

The range is all the possible outputs you can get.

graph TD
    A["Domain: Allowed Inputs"] --> B["🤖 Function"]
    B --> C["Range: Possible Outputs"]

Example: f(x) = √x (square root)

• Domain: x ≥ 0 only (can’t take square root of negative!)
• Range: y ≥ 0 (square roots are never negative)

Example: f(x) = x²

• Domain: All real numbers (square anything!)
• Range: y ≥ 0 (squares are never negative)

📏 The Vertical Line Test: Is It Really a Function?

Here’s a super easy trick to check if a graph shows a function:

Draw a vertical line anywhere on the graph.

• If it hits the graph once → ✅ It’s a function!
• If it hits the graph twice or more → ❌ NOT a function!

graph TD
    A["Draw Vertical Line"] --> B{How many times does it cross?}
    B -->|Once| C["✅ FUNCTION"]
    B -->|More than once| D["❌ NOT a function"]

Why Does This Work?

Remember: ONE input = ONE output.

A vertical line represents ONE x-value (one input). If it hits two points, that means one input is giving two outputs. That breaks the function rule!

Examples:

• Circle ❌ → vertical line hits twice
• Parabola opening up ✅ → vertical line hits once
• Wavy line (sine wave) ✅ → vertical line hits once

🧩 Piecewise Functions: The Multi-Personality Robot

Some functions have different rules for different inputs. Like a theme park with different ticket prices:

• Kids (0-12): $10
• Teens (13-17): $15
• Adults (18+): $25

Written Like This:

        ⎧ 10,  if 0 ≤ age ≤ 12
Price = ⎨ 15,  if 13 ≤ age ≤ 17
        ⎩ 25,  if age ≥ 18

Math Example:

        ⎧ x + 2,   if x < 0
f(x) =  ⎨ 5,       if x = 0
        ⎩ x²,      if x > 0

Evaluate f(-3): Since -3 < 0, use rule 1: f(-3) = -3 + 2 = -1

Evaluate f(0): Since x = 0, use rule 2: f(0) = 5

Evaluate f(2): Since 2 > 0, use rule 3: f(2) = 2² = 4

🪞 Even and Odd Functions: The Mirror Test

Even Functions = Butterfly Wings 🦋

Even functions are symmetric around the y-axis. Like a butterfly — same on both sides!

Test: f(-x) = f(x)

Example: f(x) = x²

f(3) = 9
f(-3) = 9

Same answer! ✓ It's EVEN!

Even functions: x², x⁴, cos(x), |x|

Odd Functions = Rotating Symmetry 🔄

Odd functions are symmetric around the origin. Flip it upside down AND left-right, and it looks the same!

Test: f(-x) = -f(x)

Example: f(x) = x³

f(2) = 8
f(-2) = -8

Opposite signs! ✓ It's ODD!

Odd functions: x, x³, x⁵, sin(x)

Quick Memory Trick:

• Even exponents (x², x⁴) → Even function
• Odd exponents (x¹, x³) → Odd function

⬇️⬆️ Floor and Ceiling Functions: The Rounding Robots

Floor Function ⌊x⌋: Round DOWN 📉

The floor function always rounds down to the nearest whole number below.

⌊3.7⌋ = 3    (chop off the decimal)
⌊5.2⌋ = 5    (down to 5)
⌊-2.3⌋ = -3  (careful! down means MORE negative)
⌊4⌋ = 4     (already whole, stays same)

Real life: You have $7.50 but each toy costs $3. How many can you buy? ⌊7.50 ÷ 3⌋ = ⌊2.5⌋ = 2 toys

Ceiling Function ⌈x⌉: Round UP 📈

The ceiling function always rounds up to the nearest whole number above.

⌈3.1⌉ = 4    (go up!)
⌈5.9⌉ = 6    (up to 6)
⌈-2.3⌉ = -2  (up means LESS negative)
⌈4⌉ = 4     (already whole, stays same)

Real life: You need to transport 23 people. Each car holds 5 people. ⌈23 ÷ 5⌉ = ⌈4.6⌉ = 5 cars (can’t leave anyone behind!)

🎯 Quick Summary

🚀 You Did It!

Functions are everywhere — from vending machines to video games to weather predictions. Now you understand the fundamental building blocks that power everything from your calculator to rocket science!

Remember: A function is just a dependable friend. Give it something, get back exactly one thing. Every. Single. Time. 🎉