Elementary Algebra and Functions- Core Concepts
Elementary Algebra and Functions: Core Concepts
Algebra is the bridge between arithmetic and abstract math. You learn to replace numbers with letters, find missing values, and describe relationships between quantities. Functions are a specific type of relationship — they tell you how one thing depends on another.
Most people struggle because they try to memorize steps instead of understanding what the symbols mean. This guide breaks down the actual concepts you need. No fluff.
What Algebra Actually Is
At its core, algebra is about unknowns. You have some information, something is missing, and you use rules to figure it out.
In arithmetic, you calculate: 2 + 3 = 5. In algebra, you solve: x + 3 = 5. The letter x stands for a number you don't know yet.
That's it. The rest is just learning how to manipulate equations without breaking the math.
Variables and Expressions
A variable is a placeholder — usually a letter like x, y, or n. It represents a number that can change or that you need to find.
An expression is a combination of numbers, variables, and operations. 3x + 7 is an expression. It doesn't have an equals sign, so you can't "solve" it. You can only simplify it or evaluate it if you know what x is.
Key Rules for Expressions
- Like terms can be combined.
3x + 5x = 8x.3x + 5ycannot be combined. - Use the distributive property:
a(b + c) = ab + ac. This is not optional — you'll use it constantly. - Follow the order of operations: parentheses, exponents, multiplication/division, addition/subtraction. PEMDAS isn't a suggestion.
Equations and How to Solve Them
An equation has an equals sign. It says two expressions have the same value. Your job is to find the variable that makes it true.
The golden rule: whatever you do to one side, you must do to the other. Add 5 to both sides. Divide both sides by 2. Subtract 3x from both sides. Keep it balanced.
Example: Solving a Linear Equation
Take 2x + 5 = 13.
Subtract 5 from both sides: 2x = 8.
Divide both sides by 2: x = 4.
Check it: 2(4) + 5 = 13. It works.
Types of Equations You'll Hit
- Linear equations like
ax + b = c. Graphs as a straight line. - Quadratic equations like
ax² + bx + c = 0. Graphs as a parabola. Use factoring, the quadratic formula, or completing the square. - Rational equations with fractions containing variables. Watch out for values that make the denominator zero — they break the equation.
- Systems of equations with multiple variables. Solve by substitution or elimination.
Functions: The Relationship Rule
A function is a rule that assigns exactly one output to each input. If you put in 2, you get one answer — not two, not maybe.
Written as f(x), read "f of x." It does not mean f times x. The parentheses are part of the notation.
For example: f(x) = 2x + 1.
If x = 3, then f(3) = 2(3) + 1 = 7.
Domain and Range
- The domain is all valid inputs.
- The range is all possible outputs.
For f(x) = √x, the domain is x ≥ 0 because you can't take the square root of a negative number in real algebra. The range is also y ≥ 0.
The Vertical Line Test
To check if a graph is a function, draw vertical lines across it. If any vertical line hits the graph more than once, it's not a function. That input would have two outputs, which breaks the rule.
Types of Functions You Need to Know
| Function Type | General Form | Key Feature | Graph Shape |
|---|---|---|---|
| Linear | f(x) = mx + b |
Constant rate of change | Straight line |
| Quadratic | f(x) = ax² + bx + c |
One turning point | Parabola |
| Exponential | f(x) = a(b)^x |
Grows or decays by a percentage | Curve that shoots up or down |
| Absolute Value | f(x) = |x| |
V-shape, always non-negative | Two straight lines meeting at a point |
Graphing Basics
Graphs make functions visual. The horizontal axis is usually x (input), the vertical is y or f(x) (output).
For a linear function y = mx + b:
mis the slope — how steep the line is.bis the y-intercept — where the line crosses the y-axis.
Slope formula between two points (x₁, y₁) and (x₂, y₂):
m = (y₂ - y₁) / (x₂ - x₁)
Positive slope goes uphill left to right. Negative slope goes downhill. Zero slope is flat.
Factoring and the Quadratic Formula
Factoring is rewriting an expression as a product of simpler expressions. It's the reverse of distributing.
x² + 5x + 6 = (x + 2)(x + 3)
Not every quadratic factors nicely. When it doesn't, use the quadratic formula:
x = [-b ± √(b² - 4ac)] / 2a
The part under the square root — b² - 4ac — is called the discriminant.
- If it's positive, two real solutions.
- If it's zero, one real solution.
- If it's negative, no real solutions (you'll get complex numbers).
Inequalities
Same rules as equations, with one catch: if you multiply or divide both sides by a negative number, flip the inequality sign.
-2x > 6 becomes x < -3. Flip the sign. Every time. No exceptions.
Getting Started: A Practical How-To
If you're learning this from scratch or brushing up, here's the order that actually works:
- Master integer operations. If you can't add negative numbers reliably, algebra will be painful. Fix this first.
- Learn to evaluate expressions. Plug numbers in for variables and calculate. Get comfortable with the notation.
- Solve one-step equations.
x + 4 = 9. Undo one operation. - Move to multi-step equations. Combine like terms, use distribution, isolate the variable.
- Graph linear functions. Understand slope and intercept. This connects algebra to geometry.
- Study quadratic equations. Learn to factor, then learn the quadratic formula.
- Introduce functions. Understand domain, range, and function notation. This sets you up for everything after algebra.
Don't skip steps. People who jump straight to quadratics without solid linear equation skills waste hours spinning their wheels.
Common Mistakes That Waste Your Time
- Forgetting to distribute a negative sign:
-(x - 3)is-x + 3, not-x - 3. - Adding denominators when adding fractions:
1/2 + 1/3is not2/5. Find a common denominator. - Canceling terms instead of factors. You can cancel factors in a fraction, not terms being added.
- Assuming
(a + b)² = a² + b². It doesn't. It'sa² + 2ab + b².
Where This Leads
Elementary algebra and functions are the foundation for precalculus, calculus, statistics, and most applied math. If you don't understand what a function is or how to manipulate equations, advanced math is just memorization without meaning.
Get these concepts solid. The rest builds on them.