1. **Understanding Numbers** Numbers are the basic building blocks in math. They can be classified into different types: natural numbers ($1, 2, 3, \ldots$), whole numbers ($0, 1, 2, 3, \ldots$), integers ($\ldots, -2, -1, 0, 1, 2, \ldots$), rational numbers (fractions like $\frac{3}{4}$ or decimals like $0.75$), and irrational numbers (numbers that cannot be expressed as fractions, like $\pi$ or $\sqrt{2}$). 2. **Algebra Basics** Algebra involves using symbols (usually letters) to represent numbers and express mathematical relationships. The main goal is to find the value of unknowns. - **Expressions:** Combinations of numbers, variables, and operations, e.g., $3x + 5$. - **Equations:** Statements that two expressions are equal, e.g., $3x + 5 = 11$. 3. **Solving Algebraic Equations** Example: Solve $3x + 5 = 11$. - Step 1: Subtract 5 from both sides: $3x = 11 - 5$ - Step 2: Simplify right side: $3x = 6$ - Step 3: Divide both sides by 3: $x = \frac{6}{3} = 2$ 4. **Functions** A function is a rule that assigns each input exactly one output. It is often written as $f(x)$, where $x$ is the input. - Example: $f(x) = 2x + 3$ - To find $f(4)$, substitute $x=4$: $f(4) = 2(4) + 3 = 8 + 3 = 11$ 5. **Types of Functions** - **Linear functions:** $f(x) = mx + b$, graph is a straight line. - **Quadratic functions:** $f(x) = ax^2 + bx + c$, graph is a parabola. 6. **Important Rules in Algebra and Functions** - **Distributive property:** $a(b + c) = ab + ac$ - **Combining like terms:** $3x + 4x = 7x$ - **Function evaluation:** Always substitute the input value correctly. 7. **Example: Solve and graph $f(x) = 2x - 1$** - Find $f(3)$: $2(3) - 1 = 6 - 1 = 5$ - The function is linear with slope 2 and y-intercept $-1$. This overview covers the foundational concepts of numbers, algebra, and functions with examples and step-by-step explanations to build a deeper understanding.