Lesson 4.2

Relations vs Functions

Is it reliable? That's the question. A function is a machine that never surprises you.

Introduction

In math, we connect lists of numbers. A Relation is any connection. But a Function is a special, strict relationship where every Input (x) has exactly one Output (y).

Past Knowledge

Lesson 4.1 (Coordinate Pairs). We are grouping these pairs together.

Today's Goal

Determine if a set of points or a mapping diagram represents a Function.

Future Success

This defines which graphs we can use calculus on later. It's the definition of predictability.

Key Concepts

The Vending Machine Rule

Think of as a vending machine.

Function (Working Machine)

Button A1→Chips
Button A2→Chips

OK! Different buttons can give the same snack. You know what you're getting.

Not a Function (Broken Machine)

Button B1→Candy
Button B1→Slime??

BROKEN! Pressing B1 is a gamble. One input cannot have two different results.

Worked Examples

Example 1: Mapping Diagram

Basic

Is this a function?

Input

→

Output

Check Inputs

Does any arrows split from a single input? No. 1 goes only to 5. 2 goes only to 10.

YES, it is a function.

Example 2: The Table Test

Intermediate

Is this set of points a function?

Look at X-values

We have x=1, x=3, and x=1 again.

Check Consistency

When x=1, do we get the same y? First time y=2. Second time y=5.

NO, not a function. Input 1 cheated.

Example 3: Repeating Outputs

Advanced

Is this a function?

Check Inputs

Inputs are 2, 3, 4. No repeats!

Conclusion

It doesn't matter that they all go to 7. Multiple buttons can give the same soda.

YES, it is a function.

Common Pitfalls

The "Y" Trap

Students see y-values repeating (like Example 3) and think "Not a function!" Remember: Y can repeat. X cannot.

Ordered Pair Order

5 is NOT a function because x=2 repeats. 2 IS a function because x=1 and x=5 are different.

Real-Life Applications

Passwords: A login system is a function. Input: Password → Output: Access. If you type the correct password, you should expect Access 100% of the time. If typing the same password sometimes logs you in and sometimes deletes your account, the system is broken (not a function).