Math: Algebra and Functions

TL;DR

Algebra uses letters (variables) to represent unknown numbers, helping you solve problems and find patterns. Functions are like machines that take an input, do something to it, and give you a specific output. Mastering these two areas will give you a strong foundation for more advanced math.

1. The Mental Model

Think of algebra as detective work where you're finding missing numbers using clues. Functions are like recipes: you put in ingredients (input), follow steps (the function), and get a dish (output). Each ingredient always makes the same dish.

2. The Core Material

What is Algebra?

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Algebra is essentially generalized arithmetic. Instead of just working with numbers, you use symbols (usually letters like $x$, $y$, $a$) to represent values that can change or are unknown. This allows you to write down relationships and solve for those unknowns.

Key Algebraic Concepts:

• Variables: Letters representing unknown quantities.
  • Example: In $2x + 5 = 11$, $x$ is the variable.
• Expressions: Combinations of variables, numbers, and operations (like addition, subtraction, multiplication, division) without an equals sign.
  • Example: $3y - 7$ is an expression.
• Equations: Statements that two expressions are equal, always containing an equals sign. Your goal is often to find the value(s) of the variable(s) that make the equation true.
  • Example: $4a = 20$ is an equation.
• Inequalities: Statements comparing two expressions using symbols like $<$, $>$, $\le$, $\ge$.
  • Example: $x + 3 < 10$.

Basic Algebraic Operations and Rules:

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• Combining Like Terms: You can only add or subtract terms that have the exact same variables raised to the exact same powers.
  • Example: $3x + 5x = 8x$, but $3x + 5y$ cannot be combined.
• Distributive Property: Multiply a term outside parentheses by each term inside the parentheses.
  • Example: $2(x + 3) = 2x + 6$.
• Solving Equations: The goal is to isolate the variable on one side of the equation. Whatever you do to one side, you must do to the other.
  • Addition/Subtraction Property of Equality: If $a=b$, then $a+c = b+c$ and $a-c = b-c$.
  • Multiplication/Division Property of Equality: If $a=b$ (and $c \ne 0$), then $ac = bc$ and $a/c = b/c$.

What is a Function?

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A function is a special relationship where each input has exactly one output. Think of it as a rule or a machine. We often use notation like $f(x)$ to represent a function. $f(x)$ is read as "f of x" and means "the output of the function $f$ when the input is $x$."

Key Function Concepts:

• Domain: All possible input values ($x$ values) for which the function is defined.
• Range: All possible output values ($y$ or $f(x)$ values) that the function can produce.
• Independent Variable: The input variable (usually $x$), whose value can be chosen freely.
• Dependent Variable: The output variable (usually $y$ or $f(x)$), whose value depends on the independent variable.

How to Represent Functions:

• Equation/Formula: $f(x) = 2x + 1$.
• Table of Values:
  | $x$ | $f(x)$ |
  | :-- | :----- |
  | 0 | 1 |
  | 1 | 3 |
  | 2 | 5 |
• Graph: A visual representation where inputs are on the horizontal axis and outputs on the vertical axis.
• Mapping Diagram: Shows inputs mapped to outputs with arrows.

Here's how a function works conceptually:

graph TD
    A["Input (x-value)"] --> B{"Function Rule (e.g., f(x) = 2x + 1)"};
    B --> C["Output (y or f(x) value)"];
    A -- "Example: x = 3" --> B;
    B -- "Calculation: 2(3) + 1 = 7" --> C;
    C -- "Example: Output = 7" --> D["(3, 7) on a graph"];

Evaluating Functions:

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To evaluate a function, you substitute the given input value into the function's equation wherever the variable appears.

Example: If $f(x) = x^2 - 3x + 2$, find $f(4)$.
$f(4) = (4)^2 - 3(4) + 2$
$f(4) = 16 - 12 + 2$
$f(4) = 6$

3. Worked Example

Let's solve an algebraic equation and then use the result in a function.

Problem:
1. Solve the equation: $5(x - 2) + 3x = 14$
2. Using the value of $x$ you found, evaluate the function $g(x) = x^2 - 7$.

Step-by-Step Solution:

1. Solve the equation:

   • $5(x - 2) + 3x = 14$
   • Apply the distributive property: $5x - 10 + 3x = 14$
   • Combine like terms: $(5x + 3x) - 10 = 14$
   • $8x - 10 = 14$
   • Add 10 to both sides: $8x - 10 + 10 = 14 + 10$
   • $8x = 24$
   • Divide both sides by 8: $\frac{8x}{8} = \frac{24}{8}$
   • $x = 3$

2. Evaluate the function $g(x) = x^2 - 7$ using $x=3$:

   • $g(3) = (3)^2 - 7$
   • $g(3) = 9 - 7$
   • $g(3) = 2$

So, the solution to the equation is $x=3$, and when you put $3$ into the function $g(x)$, you get $2$.

4. Key Takeaways

• Algebra uses variables to represent unknowns, helping you generalize math problems.
• Equations state two expressions are equal; solving them means finding the variable's value that makes the statement true.
• Functions are rules where each input produces exactly one output, like a reliable machine.
• Understanding domain (possible inputs) and range (possible outputs) is crucial for functions.
• To evaluate a function, substitute the input value into the function's formula.

Common Mistakes to Avoid:

• Not distributing correctly: Remember to multiply the term outside parentheses by every term inside.
• Forgetting to do the same to both sides of an equation: This throws off the balance and leads to incorrect answers.
• Combining unlike terms: You can't add apples and oranges (e.g., $3x + 2y$ stays as is).
• Confusing $f(x)$ with "f times x": It's "f of x," meaning the function $f$ applied to the input $x$.

5. Now Try It

Spend 15 minutes trying to solve this problem:

1. Solve for $y$: $2(y + 4) - 5y = 17$
2. Using the value of $y$ you found, find $h(y)$ for the function $h(y) = 3y + 10$.

What success looks like: You should get a specific number for $y$ and a specific number for $h(y)$, showing you can handle both algebraic solving and function evaluation.