Functions are an important idea in pre-calculus, and knowing their main features can really help you with math. Let’s break these features down in simple terms:

1. What is a Function?

A function is a way to connect inputs to outputs. Each input gives you exactly one output. For example, the function (f(x) = 2x + 3) gives each value of (x) a special output. So, if you put in (1), you get (f(1) = 5).

2. Domain and Range

• Domain: This is all the possible input values (often called (x) values). For example, in (f(x) = \sqrt{x}), the domain is (x \geq 0) because you can’t find the square root of a negative number.

• Range: This is all the possible output values (often called (f(x)) values). For (f(x) = x^2), the range is (y \geq 0), meaning you can only get zero or positive numbers.

3. Types of Functions

Functions can be grouped by their types:

• Linear Functions: They look like (f(x) = mx + b) (for example, (f(x) = 2x + 1)).
• Quadratic Functions: They take the shape of (f(x) = ax^2 + bx + c) (like (f(x) = x^2 - 4)).
• Exponential Functions: These are in the form (f(x) = a \cdot b^x) (like (f(x) = 2^x)).

4. Even and Odd Functions

Functions can also be divided into two main types:

• Even Functions: These look the same on both sides of the (y)-axis (like (f(x) = x^2)).
• Odd Functions: These look the same if you turn them around the origin (like (f(x) = x^3)).

By understanding these key ideas, you'll have a good base for working with functions in pre-calculus!