🎯 Function Foundations: Your Journey Begins!

Imagine you have a magic machine. You put something in, the machine does its magic, and something comes out. That’s exactly what a function is in math!

🤖 What is a Function?

Think of a function like a vending machine:

• You put in money (the input)
• The machine processes it
• You get a snack (the output)

The Rule: For every input, there’s exactly ONE output. You can’t put in $1 and get both chips AND candy at the same time!

Simple Example

f(x) = x + 2

This says: “Whatever number you give me, I’ll add 2 to it!”

Real Life Example:

• Your age function: Every year, you add 1 to your age
• If you’re 8 now, next year you’ll be 9!

📊 Functions and Their Graphs

A graph is like a treasure map that shows you where every input leads to its output!

Drawing Our Machine’s Story

When we draw f(x) = x + 2:

• Every point on the line is an (input, output) pair
• The x-axis shows inputs (what you put in)
• The y-axis shows outputs (what you get)

graph TD
    A["Pick a number x"] --> B["Put it in function"]
    B --> C["Get output y"]
    C --> D["Plot point x,y"]
    D --> E["Connect all points"]
    E --> F["You have a graph!"]

The Vertical Line Test 🎯

How do you know if a picture is a function? Use the vertical line test!

The Rule: Draw a vertical line anywhere. If it touches the graph in more than one place, it’s NOT a function!

Why? Because one input would give two outputs. That breaks our vending machine rule!

🎪 Domain and Range: The Playground Rules

Domain: “What CAN I Put In?”

The domain is like the menu at a restaurant. It tells you what inputs are allowed!

Think of it this way:

• You can’t divide by zero (the machine breaks!)
• You can’t take a square root of negative numbers (with real numbers)

Example 1: f(x) = 1/x

❌ Can’t use x = 0 (dividing by zero!) ✅ Everything else works! Domain: All numbers except 0

Example 2: f(x) = √x

❌ Can’t use negative numbers (no real square root!) ✅ Zero and positive numbers work! Domain: x ≥ 0 (zero and up)

Range: “What CAN I Get Out?”

The range is like the prizes at a carnival game. It tells you all possible outputs!

Example: f(x) = x²

Notice: Outputs are always 0 or positive! Range: y ≥ 0

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

🎮 Quick Memory Trick

• Domain = Door (what can come IN)
• Range = Result (what can come OUT)

🔗 Function Composition: Connecting Machines!

What if you have TWO vending machines in a row?

The output from the first machine becomes the input for the second!

The Story

Imagine:

1. Machine f: Doubles your number → f(x) = 2x
2. Machine g: Adds 3 → g(x) = x + 3

If you connect them: Put 5 into machine f, take what comes out, put it into machine g!

Start with 5
↓
Machine f: 2 × 5 = 10
↓
Machine g: 10 + 3 = 13
↓
Final answer: 13

Writing It Down

We write this as: (g ∘ f)(x) or g(f(x))

Read it as: “g of f of x” — f goes first, then g!

Step-by-Step Example

Let f(x) = 2x and g(x) = x + 3

Find (g ∘ f)(5):

1. First, find f(5) = 2 × 5 = 10
2. Then, find g(10) = 10 + 3 = 13
3. Answer: 13

Find (f ∘ g)(5):

1. First, find g(5) = 5 + 3 = 8
2. Then, find f(8) = 2 × 8 = 16
3. Answer: 16

⚠️ Important Discovery!

Order matters! (g ∘ f)(5) = 13 but (f ∘ g)(5) = 16

Just like putting on socks then shoes is different from shoes then socks! 🧦👟

graph LR
    A["Input x"] --> B["f: first function"]
    B --> C["Result"]
    C --> D["g: second function"]
    D --> E["Final Output"]

🔄 Inverse Functions: The Undo Button!

What if your magic machine could reverse itself?

An inverse function is like pressing CTRL+Z — it undoes what the original function did!

The Story

Machine f: Adds 5 → f(x) = x + 5

• Put in 3, get 8

Machine f⁻¹ (f inverse): Subtracts 5 → f⁻¹(x) = x - 5

• Put in 8, get back 3!

You’re back where you started!

How to Find an Inverse

Let’s find the inverse of f(x) = 2x + 3

Step 1: Replace f(x) with y

y = 2x + 3

Step 2: Swap x and y

x = 2y + 3

Step 3: Solve for y

x - 3 = 2y
y = (x - 3)/2

Step 4: Write as f⁻¹(x)

f⁻¹(x) = (x - 3)/2

Testing Our Inverse ✅

If f(4) = 2(4) + 3 = 11

Then f⁻¹(11) = (11 - 3)/2 = 4 ✓

We got back to 4! The inverse works!

graph LR
    A["4"] -->|f: multiply by 2, add 3| B["11"]
    B -->|f⁻¹: subtract 3, divide by 2| A

The Mirror Test: Graph Check

The graph of f and f⁻¹ are mirror images across the line y = x!

If you fold the paper along y = x, the two graphs would match up perfectly!

⚠️ Not All Functions Have Inverses!

Only one-to-one functions have inverses!

A function is one-to-one if each output comes from only ONE input.

Test: Use the horizontal line test

• If any horizontal line crosses the graph more than once → No inverse!

🎉 You Did It!

You now understand the foundations of functions:

✅ Functions are input-output machines with one output per input

✅ Graphs show all input-output pairs as points

✅ Domain is what you CAN put in

✅ Range is what you CAN get out

✅ Composition connects functions like machines in a row

✅ Inverse functions undo what the original function did

🧠 Key Formulas to Remember

You’re ready for the next adventure in calculus! 🚀