Math Antics: Algebra Basics - Functions

Introduction to Functions

• Definition of Function in Math:

  • Relates one set to another set.
  • A set is a collection of things (often numbers, but can be other items).

• Visual Representation of Sets:

  • Sets are often shown using curly brackets {} with elements separated by commas.
  • Sets can be finite (e.g., alphabet letters) or infinite (e.g., integers).

How Functions Work

• Input and Output Sets:

  • Input Set (Domain): The set from which values are taken.
  • Output Set (Range): The set to which values are mapped.

• Function Table:

  • Consists of two columns: inputs on the left, outputs on the right.
  • Function shown above the table as a mathematical rule.

• Example:

  • Input set: {triangle, square, pentagon...}
  • Function: "Output the number of sides."
  • Each input relates to a specific output (e.g., triangle to 3).

Algebraic Functions

• Simple Algebraic Function Example:

  • Equation: y = 2x
  • Input set (domain): values of x
  • Output set (range): resulting y values
  • Function table example for x = 1, 2, 3: outputs y = 2, 4, 6.

• Key Rule for Functions:

  • Each input must produce exactly one output (no "one-to-many" relations).
  • Example of disqualified function: y^2 = x (produces two outputs for one input).

Graphing Functions

• Linear Function Example:

  • Equation: y = x + 1
  • Graph can be represented as ordered pairs on a coordinate plane.
  • Forms a straight line known as a "linear function."

• Vertical Line Test:

  • Determines if a graph represents a function.
  • A graph passes if a vertical line intersects it at only one point for each input x.

Function Notation

• Function Notation:

  • Example: f(x) = y
  • Represents a function named f with input x and output y.
  • Avoids misinterpretation as multiplication and emphasizes function relationships.

• Evaluating Functions:

  • Substitute specific values into the function notation (e.g., finding f(4) in f(x) = 3x + 2).
  • Results in specific output values (e.g., f(4) = 14).

Summary

• Functions relate input values to exactly one output value.
• Domain: set of all possible input values.
• Range: set of all possible output values.
• Functions in algebra are often equations that can be graphed as ordered pairs on a coordinate plane.

---

• Further Learning:
  • Practice exercises to reinforce understanding of functions.
  • Visit Math Antics for more resources.