Function Operations

Function Operations: The Ultimate Function Toolkit 🧰

Imagine functions are like magic machines. You put something in, and something else comes out. Now, what if you could combine these machines? Stack them? Make them work backwards? That’s what function operations are all about!

🎪 The Big Picture: Functions as Team Players

Think of functions like workers in a factory. Each worker (function) has a specific job. But when workers team up, they can do amazing things together!

Our analogy throughout: Functions are like recipe instructions. Adding recipes together? Combining them? Running them in sequence? That’s function operations!

➕ Function Addition: Mixing Two Recipes

When you add two functions, you’re saying: “Do both jobs and add the results!”

How It Works

If you have two functions:

• f(x) = 2x (doubles everything)
• g(x) = 3 (always gives 3)

Then (f + g)(x) = f(x) + g(x) = 2x + 3

Real Example

f(x) = x² and g(x) = 5x

(f + g)(x) = x² + 5x

When x = 2:

• f(2) = 4
• g(2) = 10
• (f + g)(2) = 4 + 10 = 14 ✓

graph TD
    A["Input: x"] --> B["f of x"]
    A --> C["g of x"]
    B --> D["Add Results"]
    C --> D
    D --> E["f + g output"]

💡 Think of it like this: You ask two friends to count apples. Friend A counts 4, Friend B counts 10. Together they counted 14!

➖ Function Subtraction: Finding the Difference

Subtraction works the same way—just subtract instead of add!

The Formula

(f - g)(x) = f(x) - g(x)

Real Example

f(x) = 5x and g(x) = 2x

(f - g)(x) = 5x - 2x = 3x

When x = 4:

• f(4) = 20
• g(4) = 8
• (f - g)(4) = 20 - 8 = 12 ✓

🎯 Pro tip: Subtraction tells you how much MORE one function gives compared to another!

✖️ Function Multiplication: Super Combo Power

When you multiply functions, you multiply their outputs!

The Formula

(f · g)(x) = f(x) × g(x)

Real Example

f(x) = x and g(x) = x + 1

(f · g)(x) = x × (x + 1) = x² + x

When x = 3:

• f(3) = 3
• g(3) = 4
• (f · g)(3) = 3 × 4 = 12 ✓

🌟 Imagine: One machine makes boxes (x), another adds a bonus box (x+1). Multiply them? You get a grid of x² + x boxes!

🔗 Composition of Functions: The Assembly Line

This is where it gets exciting! Composition means feeding one function’s output into another function.

The Magic Notation

(f ∘ g)(x) = f(g(x))

Read it as: “f of g of x” — First do g, then take that answer and feed it to f!

Real Example

f(x) = x² and g(x) = x + 1

(f ∘ g)(x) = f(g(x)) = f(x + 1) = (x + 1)²

When x = 2:

1. First: g(2) = 2 + 1 = 3
2. Then: f(3) = 3² = 9 ✓

graph TD
    A["Input: x"] --> B["g: add 1"]
    B --> C["Result: x+1"]
    C --> D["f: square it"]
    D --> E["Output: x+1 squared"]

⚠️ Order Matters!

(f ∘ g)(x) ≠ (g ∘ f)(x) usually!

• (f ∘ g)(2) = f(g(2)) = f(3) = 9
• (g ∘ f)(2) = g(f(2)) = g(4) = 5

Different answers! It’s like putting on socks then shoes vs. shoes then socks! 🧦👟

🔄 Inverse Functions: The Undo Button

An inverse function reverses what the original function did!

The Key Idea

If f takes you from A to B, then f⁻¹ (f-inverse) takes you from B back to A.

The Test

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Real Example

f(x) = 2x (doubles everything)

f⁻¹(x) = x/2 (halves everything)

Check: f(f⁻¹(6)) = f(3) = 6 ✓

graph LR
    A["5"] -->|f: double| B["10"]
    B -->|f inverse: halve| A

Finding an Inverse (Step by Step)

1. Write y = f(x)
2. Swap x and y
3. Solve for y
4. That’s your f⁻¹(x)!

Example: f(x) = 3x + 2

1. y = 3x + 2
2. x = 3y + 2
3. x - 2 = 3y → y = (x - 2)/3
4. f⁻¹(x) = (x - 2)/3 ✓

🎯 One-to-One Functions: No Twins Allowed!

A function is one-to-one if every output comes from exactly ONE input.

The Rule

If f(a) = f(b), then a = b

No two different inputs give the same output!

Visual Test: Horizontal Line Test

Draw horizontal lines across the graph. If any line crosses more than once, it’s NOT one-to-one!

Why It Matters

Only one-to-one functions have inverses! If twins exist, the inverse wouldn’t know which twin to go back to!

graph TD
    subgraph One-to-One
    A1["1"] --> B1["2"]
    A2["2"] --> B2["4"]
    A3["3"] --> B3["6"]
    end

🎨 Onto Functions (Surjective): Everything Gets Hit!

A function is onto if every possible output value is actually produced by some input.

The Idea

No output is left behind! Every element in the target set has an arrow pointing to it.

Real Example

f: {1,2,3} → {a,b} where f(1)=a, f(2)=b, f(3)=a

Is this onto? Yes! Both ‘a’ and ‘b’ are hit.

g: {1,2} → {a,b,c} where g(1)=a, g(2)=b

Is this onto? No! ‘c’ is never reached!

🎪 Think of it like: A bus route. An “onto” route stops at EVERY station. No station is skipped!

🏆 Bijective Functions: The Perfect Match!

A function is bijective if it’s BOTH:

1. ✅ One-to-One (no twins)
2. ✅ Onto (everything gets hit)

Why Bijections Are Special

Bijective functions are perfectly invertible! Every input matches exactly one output, and vice versa.

Real Example

f: {1,2,3} → {a,b,c}

f(1) = a, f(2) = b, f(3) = c

• One-to-One? ✅ Each output comes from one input
• Onto? ✅ a, b, c are all covered
• Bijective? ✅ YES!

graph LR
    subgraph Domain
    D1["1"]
    D2["2"]
    D3["3"]
    end
    subgraph Range
    R1["a"]
    R2["b"]
    R3["c"]
    end
    D1 --> R1
    D2 --> R2
    D3 --> R3

📈 Increasing Functions: Always Going Up!

A function is increasing if bigger inputs give bigger outputs.

The Rule

If a < b, then f(a) < f(b)

As you walk right on the graph, you always climb up!

Examples

Real Example

f(x) = 3x + 1

• f(1) = 4
• f(2) = 7
• f(3) = 10

Each step up in x → step up in f(x). Increasing! 📈

📉 Decreasing Functions: Sliding Down!

A function is decreasing if bigger inputs give SMALLER outputs.

The Rule

If a < b, then f(a) > f(b)

As you walk right on the graph, you always slide down!

Examples

Real Example

f(x) = 10 - x

• f(1) = 9
• f(2) = 8
• f(3) = 7

Each step up in x → step DOWN in f(x). Decreasing! 📉

🎓 Summary: Your Function Operations Toolkit

🌟 You did it! You now have the complete toolkit for combining, reversing, and analyzing functions. These operations are the building blocks for calculus, computer science, and beyond. Keep practicing—you’ve got this!