Functions in Math

Functions

Definition

• A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
• A function is a way of describing a relationship between variables.

Notation

• Functions are often denoted by letters such as f, g, or h.
• The function notation is f(x) = output, where x is the input.

Domain and Range

• The domain is the set of all possible inputs (x-values) of a function.
• The range is the set of all possible outputs (y-values) of a function.

Types of Functions

• Linear Functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
• Quadratic Functions: f(x) = ax^2 + bx + c, where a, b, and c are constants.
• Exponential Functions: f(x) = a^x, where a is a constant.
• Polynomial Functions: f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where n is a positive integer.

Function Operations

• Addition: (f + g)(x) = f(x) + g(x)
• Subtraction: (f - g)(x) = f(x) - g(x)
• Multiplication: (f × g)(x) = f(x) × g(x)
• Composition: (f ∘ g)(x) = f(g(x))

Graphing Functions

• The graph of a function is a visual representation of the relationship between the input and output.
• The x-axis represents the domain, and the y-axis represents the range.
• The graph can be used to identify key features such as the x-intercept, y-intercept, and vertex.

Function Properties

• Even and Odd Functions:
  • Even function: f(-x) = f(x)
  • Odd function: f(-x) = -f(x)
• Increasing and Decreasing Functions:
  • Increasing function: f(x) ≤ f(y) if x ≤ y
  • Decreasing function: f(x) ≥ f(y) if x ≤ y

Functions

Definition

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range), describing a relationship between variables.

Notation and Basics

• Functions are denoted by letters such as f, g, or h.
• Function notation is f(x) = output, where x is the input.
• Domain is the set of all possible inputs (x-values) of a function.
• Range is the set of all possible outputs (y-values) of a function.

Types of Functions

Linear Functions

• f(x) = mx + b, where m is the slope and b is the y-intercept.

Quadratic Functions

• f(x) = ax^2 + bx + c, where a, b, and c are constants.

Exponential Functions

• f(x) = a^x, where a is a constant.

Polynomial Functions

• f(x) = a_nx^n + a_(n-1)x^(n-1) +...+ a_1x + a_0, where n is a positive integer.

Function Operations

• (f + g)(x) = f(x) + g(x) (Addition)
• (f - g)(x) = f(x) - g(x) (Subtraction)
• (f × g)(x) = f(x) × g(x) (Multiplication)
• (f ∘ g)(x) = f(g(x)) (Composition)

Graphing Functions

• Graph of a function is a visual representation of the relationship between input and output.
• x-axis represents the domain, and y-axis represents the range.
• Graph can be used to identify key features such as x-intercept, y-intercept, and vertex.

Function Properties

Even and Odd Functions

• Even function: f(-x) = f(x)
• Odd function: f(-x) = -f(x)

Increasing and Decreasing Functions

• Increasing function: f(x) ≤ f(y) if x ≤ y
• Decreasing function: f(x) ≥ f(y) if x ≤ y