Functions, Relations and Basic Algebra

TL;DR

You'll learn about relations, which are simply connections between sets of values, and functions, which are special types of relations where each input has exactly one output. We'll also cover basic algebraic operations you'll need to manipulate these expressions.

1. The Mental Model

Think of relations as a list of ordered pairs, like (input, output). Functions are a stricter version of this, where for any given input, there's only one specific output. Algebra gives you the tools to change or understand these relationships.

2. The Core Material

Relations: The General Connection

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A relation is just a set of ordered pairs, usually written as (x, y). The first element (x) is from the domain (all possible inputs), and the second element (y) is from the range (all possible outputs).
For example, {(1, 2), (1, 3), (2, 4)} is a relation. Here, 1 maps to both 2 and 3.

Functions: The Unique Output Rule

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A function is a special type of relation where each element in the domain maps to exactly one element in the range. This is the crucial difference. If you have an input, you know precisely what the output will be.
Using the example above, {(1, 2), (1, 3), (2, 4)} is not a function because 1 maps to both 2 and 3.
However, {(1, 2), (2, 3), (3, 4)} is a function. Each input has only one output.

We often write functions using f(x) notation, where f(x) represents the output (which is y) for a given input x. So, y = x + 2 can also be written as f(x) = x + 2.

graph TD
    subgraph "Is it a Function?"
        A("Start with a Relation") --> B{Does each input (x)
        have ONLY ONE output (y)?}
        B -- "Yes" --> C["It's a Function!"]
        B -- "No" --> D["It's NOT a Function (just a Relation)"]
    end

Basic Algebra: Manipulating Expressions

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Algebra provides the rules for working with variables, numbers, and operations. This is fundamental for evaluating functions, solving for unknowns, and simplifying expressions.

Order of Operations (PEMDAS/BODMAS)

Remember the order:
1. Parentheses (or Brackets)
2. Exponents (or Orders)
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Solving Equations

The goal is usually to isolate a variable. Whatever you do to one side of the equation, you must do to the other.
* x + 5 = 10 => Subtract 5 from both sides: x = 5
* 3x = 12 => Divide by 3 on both sides: x = 4
* x / 2 = 7 => Multiply by 2 on both sides: x = 14

Simplifying Expressions

Combine like terms:
* 3x + 2y - x + 5y => (3x - x) + (2y + 5y) => 2x + 7y
Distribute:
* 2(x + 3) => 2x + 6

3. Worked Example

Let's say you have the relation R = {(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)}.
1. Is R a function?
* Look at each input (x value):
* -2 maps to 4 (unique)
* -1 maps to 1 (unique)
* 0 maps to 0 (unique)
* 1 maps to 1 (unique)
* 2 maps to 4 (unique)
* Since every input x has exactly one output y, yes, R is a function.

2. If R is a function, write it using function notation and evaluate f(3) if f(x) = x^2.
   • Based on the pairs, it looks like y = x^2. So, we can write f(x) = x^2.
   • To evaluate f(3), you substitute 3 for x in the function's rule:
     f(3) = (3)^2
f(3) = 9

4. Key Takeaways

• A relation is any set of ordered pairs connecting inputs to outputs.
• A function is a special relation where each input has exactly one output.
• The vertical line test is a quick visual way to check if a graph represents a function.
• f(x) notation means "the output of the function f when the input is x."
• Always follow the order of operations (PEMDAS/BODMAS) when evaluating expressions.
• To solve equations, perform the same inverse operation on both sides to isolate the variable.
• You combine like terms and distribute to simplify algebraic expressions.

Common Mistakes to Avoid:
- Confusing relations with functions; remember the "one output per input" rule for functions.
- Incorrectly applying the order of operations, especially with multiplication/division and addition/subtraction.
- Forgetting to apply operations to both sides of an equation when solving.
- Mixing up variables when simplifying (e.g., trying to combine 3x and 2y).

5. Now Try It

Given the equation 2x - 3y = 6, can y be expressed as a function of x? If so, rewrite the equation in the form y = f(x). Then, find f(0) and f(3).

Success looks like: You correctly isolate y to show y is a function of x, and you correctly substitute 0 and 3 into your new function to get their respective output values.