Getting Started with General Mathematics

Mathematics surrounds us every day, from simple equations like y = x² to complex formulas like E = mc². These expressions might look intimidating, but they're just tools that help us understand patterns and solve real problems.

General Mathematics builds on what you already know and introduces you to more powerful concepts. You'll work with angles (like 90°), constants (like π), and various mathematical operations that connect to everything from physics to economics.

Quick Tip: Don't worry if some symbols look unfamiliar - you'll master them step by step!

Understanding Functions and Variables

A function is basically a mathematical rule where each input (domain) connects to exactly one output (range). Think of it like a vending machine - you press one button (input) and get exactly one specific snack (output).

The domain (x) is your independent variable because you can choose any value for it. The range (y) is your dependent variable because its value depends completely on what you chose for x. For example, in y = x², when x = 2, you get y = 4, and when x = 3, you get y = 9.

This "one input, one output" rule is what makes functions so reliable and useful in solving problems. You can always predict what you'll get!

Remember: If one input gives you multiple outputs, it's not a function - it's just a relation.

Five Ways to Show Mathematical Relationships

You can represent the same function in five different ways, and each method has its advantages. A mapping diagram shows arrows connecting inputs to outputs, while a table of values lists x and y pairs neatly in columns.

Set of ordered pairs write relationships as coordinates like (2,4) and (3,9). Graphs give you a visual picture of how the function behaves, making patterns easy to spot.

The function machine concept helps you visualize the process - imagine x = 2 going into a machine and y = 4 coming out. All five methods show the same mathematical relationship, just in different formats.

Study Smart: Practice switching between these representations - it's a common test question!

Relations vs Functions: What's the Difference?

A relation connects domain values to range values, but it doesn't have to follow the "one input, one output" rule. For example, when x = 4 in y = √x, you get both y = 2 and y = -2, creating ordered pairs (4,2) and (4,-2).

A function is stricter - it's a special type of relation where each domain element connects to only one range value. In y = x² + 1, when x = 1, you get exactly one answer: y = 2, giving you the ordered pair (1,2).

The key difference? Functions are predictable and reliable, while relations can have multiple outputs for the same input. This makes functions incredibly useful for modeling real-world situations.

Test Tip: If you see the same x-value paired with different y-values, it's a relation but not a function!

Domain, Range, and Asymptotes

Domain and range tell you the boundaries of your function. Domain shows all possible input values, while range shows all possible output values. You'll often see these written as D = (-∞,∞) or R = (0,∞).

Horizontal asymptotes are invisible horizontal lines that graphs approach but never actually touch. Similarly, vertical asymptotes are vertical lines that graphs get infinitely close to but never cross.

Understanding these concepts helps you predict how functions behave at their extreme values. Some functions work for all real numbers, while others have restrictions that create interesting boundary behaviors.

Visual Learner? Sketch the graphs to see how asymptotes create "forbidden zones" that functions can't enter.

Domain and Range: The Complete Picture

The domain represents your independent variable - it's your input, x-variable, and first coordinate all rolled into one. The range represents your dependent variable - your output, y-variable, and second coordinate.

When you see notation like D = (-∞,∞), it means the domain includes all real numbers from negative infinity to positive infinity. For range like R = {y | y ≥ 2}, you're looking at all y-values greater than or equal to 2.

Infinity (∞) doesn't have a defined value because it represents an endless concept. The negative direction (-∞) points toward negative values, while positive infinity (∞) extends toward positive values.

Remember: Square brackets $$ include the endpoint, while parentheses ( ) exclude it!

Finding Domain: Three Key Rules

Finding the domain of a function follows three main rules. If y is a polynomial, your domain is all real numbers - no restrictions! If y is a rational expression with a denominator d(x), exclude any x-values that make d(x) = 0.

For radical expressions like √r(x) where n is even, your domain includes only x-values that make r(x) greater than or equal to zero. You can't take the square root of negative numbers in real mathematics.

These rules help you identify where functions are defined and where they might have problems. Understanding domain restrictions prevents mathematical errors and helps you interpret graphs correctly.

Quick Check: Always look for denominators that could equal zero or even roots of negative numbers!

Finding Range: Mirror the Process

Finding range works similarly to finding domain, but you solve for x in terms of y instead. If x becomes a polynomial expression, your range is all real numbers (minus any restrictions from the original equation).

When x becomes a rational expression with denominator d(y), exclude y-values that make d(y) equal zero. For radical expressions involving y, include only values that keep the expression under the radical non-negative.

Practice with examples like 3x - y = 4: solving for y gives you y = 3x - 4 (domain: all real numbers), while solving for x gives you x = $y + 4$/3 (range: all real numbers).

Pro Tip: Always check your work by substituting a few values back into the original equation!

Working with Polynomials and Evaluating Functions

Polynomial functions like y = x² - 4 have domains of all real numbers since polynomials are defined everywhere. For the range, solve for x: x = √$y + 4$, which means y ≥ -4.

Evaluating functions means substituting specific values. For f(x) = 2x + 1, finding f(2) means replacing x with 2: f(2) = 2(2) + 1 = 5. It's like following a recipe with specific ingredients.

This process helps you find specific points on graphs and solve real-world problems. Whether you're calculating costs, distances, or growth rates, function evaluation gives you concrete answers.

Practice Makes Perfect: Try evaluating functions with different types of numbers - positive, negative, and fractions!

Function Evaluation Practice Problems

Let's work through some examples to build your confidence. For f(x) = x² + 3x - 2 and f(-3): substitute -3 for every x to get f(-3) = (-3)² + 3(-3) - 2 = 9 - 9 - 2 = -2.

For g(x) = 3x - 4 and g(-2): substitute -2 to get g(-2) = 3(-2) - 4 = -6 - 4 = -10. The process stays the same regardless of the function's complexity.

More complex functions like h(t) = t² + 2t + 3 work the same way. Just substitute your given value carefully and follow the order of operations. Double-check your arithmetic to avoid simple mistakes.

Success Strategy: Work step-by-step and show all your work - partial credit counts, and you'll catch errors easier!