Introduction to Functions

Functions

Definition:

• A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
• It is denoted by a letter such as f, g, or h, and is read as "f of x" or "f(x)".
• A function can be thought of as a machine that takes an input (or inputs) and produces a corresponding output.

Key Concepts:

• Domain: The set of all possible inputs (x-values) of a function.
• Range: The set of all possible outputs (y-values) of a function.
• Function notation: A shorthand way of writing functions, e.g., f(x) = x^2.

Types of Functions:

• Linear functions: Functions with a constant rate of change, e.g., f(x) = 2x + 3.
• Quadratic functions: Functions of the form f(x) = ax^2 + bx + c, where a ≠ 0.
• Exponential functions: Functions of the form f(x) = a^x, where a is a constant.
• Polynomial functions: Functions consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Algebraic Expressions

Simplifying Expressions:

• Combining like terms: Combining terms with the same variable(s) and coefficient(s).
• Distributive property: Expanding products of sums using the rule a(b + c) = ab + ac.

Algebraic Properties:

• Commutative property of addition: a + b = b + a
• Associative property of addition: (a + b) + c = a + (b + c)
• Distributive property: a(b + c) = ab + ac

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
• A function can be thought of as a machine that takes an input (or inputs) and produces a corresponding output.
• Functions are denoted by a letter (e.g. f, g, or h) and are read as "f of x" or "f(x)".

Key Concepts

• Domain: The set of all possible inputs (x-values) of a function.
• Range: The set of all possible outputs (y-values) of a function.
• Function notation: A shorthand way of writing functions, e.g., f(x) = x^2.

Types of Functions

• Linear functions: Functions with a constant rate of change, e.g., f(x) = 2x + 3.
• Quadratic functions: Functions of the form f(x) = ax^2 + bx + c, where a ≠ 0.
• Exponential functions: Functions of the form f(x) = a^x, where a is a constant.
• Polynomial functions: Functions consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Algebraic Expressions

• Simplifying Expressions: Combining like terms by combining terms with the same variable(s) and coefficient(s).
• Distributive property: Expanding products of sums using the rule a(b + c) = ab + ac.

Algebraic Properties

• Commutative property of addition: a + b = b + a.
• Associative property of addition: (a + b) + c = a + (b + c).
• Distributive property: a(b + c) = ab + ac.