Foundational Algebra and Functions

TL;DR

Algebra is like a language for solving puzzles, using symbols to represent unknown numbers. Functions are special kinds of relationships where every input has exactly one output. Together, they're super powerful tools for modeling real-world situations.

1. The Mental Model

Think of algebra as figuring out what numbers you need to put into a recipe to get a specific outcome. Functions are like well-behaved machines: you put something in, and you always get the same, single thing out.

2. The Core Material

Variables and Expressions

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In algebra, we use letters (like x, y, t) to represent numbers we don't know yet, or numbers that can change. These are called variables. An expression is a combination of variables, numbers, and operations (like +, -, *, /). It's a mathematical phrase, not a complete sentence.

• x + 5 (Here, x is a variable, 5 is a constant, and + is an operation.)
• 3y - 7
• 2ab (This means 2 * a * b)

Equations

An equation is like a complete mathematical sentence. It says that two expressions are equal. Our goal is often to solve an equation, meaning to find the value(s) of the variable(s) that make the equation true.

x + 5 = 10
To solve for x, we want to get x by itself. We do the same thing to both sides of the equation to keep it balanced:
x + 5 - 5 = 10 - 5
x = 5

Inequalities

An inequality is similar to an equation, but instead of showing equality, it shows a relationship of "greater than," "less than," etc.
* > (greater than)
* < (less than)
* >= (greater than or equal to)
* <= (less than or equal to)

Solving inequalities is much like solving equations, with one crucial difference: if you multiply or divide both sides by a negative number, you must flip the inequality sign.

2x - 3 < 7
2x < 10
x < 5

-3x + 1 >= 10
-3x >= 9
x <= -3 (Notice the sign flip because we divided by -3!)

What's a Function?

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A function is a rule that assigns exactly one output to each input. Think of it as a special kind of relationship. We often write functions using "f(x)" notation, which you read as "f of x." Here, x is the input, and f(x) is the output.

• f(x) = 2x + 1 (For any x you put in, you get one 2x+1 out.)
• If x = 3, then f(3) = 2(3) + 1 = 7.
• If x = -1, then f(-1) = 2(-1) + 1 = -1.

The set of all possible inputs is called the domain. The set of all possible outputs is called the range.

graph TD
    A["Input (x)"] --> B["Function Rule (e.g., f(x) = 2x + 1)"];
    B --> C["Unique Output (f(x))"];
    C --> D["Each 'x' maps to only one 'f(x)'"];

Types of Functions

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You'll encounter many types, but some common ones are:

• Linear Functions: f(x) = mx + b. When you graph these, you get a straight line. m is the slope (steepness), and b is the y-intercept (where it crosses the vertical axis).
• Quadratic Functions: f(x) = ax^2 + bx + c. These graphs are parabolas (U-shapes).
• Polynomial Functions: Generalizations of linear and quadratic functions, involving sums of terms with x raised to non-negative integer powers (e.g., x^3 + 2x - 5).

3. Worked Example

Let's work through an example involving an equation and then evaluate a function.

Problem: You're saving up for a new game that costs $60. You already have $15, and you save $5 each week.
1. Write an equation to represent how many weeks (w) it will take to save enough money.
2. Solve the equation.
3. Let C(w) be a function representing the total money you have after w weeks. Evaluate C(4).

Solution:

1. Equation:
   Your starting money is $15.
   You save $5 per week, so after w weeks, you'll have 5w more.
   You want the total to be $60.
   Equation: 15 + 5w = 60

2. Solve the equation:
   15 + 5w = 60
   Subtract 15 from both sides:
   5w = 60 - 15
5w = 45
   Divide by 5:
   w = 45 / 5
w = 9
   It will take 9 weeks to save enough money.

3. Evaluate the function C(4):
   First, define the function: C(w) = 15 + 5w.
   To evaluate C(4), substitute w = 4 into the function:
   C(4) = 15 + 5(4)
C(4) = 15 + 20
C(4) = 35
   After 4 weeks, you'll have