A function in mathematics is a relation between a set of inputs and a set of possible outputs, where each input is related to exactly one output.

In formal terms, a function $f$ from a set $A$ to a set $B$ is denoted as:

$f: A \to B$

This means that for every element $x \in A$, there is a unique element $y \in B$ such that $y = f(x)$. In other words, the function assigns one output for each input.

Key characteristics of functions:

1. Domain: The set of all possible inputs (the set $A$).
2. Range: The set of all possible outputs (the set $B$).
3. Rule of correspondence: The specific rule that assigns each input in the domain to exactly one output in the range.

Example:

Consider a function $f(x) = 2x + 3$. For each value of $x$, there is a unique corresponding output $y$:

• If $x = 1$, then $f(1) = 2(1) + 3 = 5$.
• If $x = 2$, then $f(2) = 2(2) + 3 = 7$.

Types of Functions:

• Linear functions: Functions of the form $f(x) = mx + b$, where $m$ and $b$ are constants.
• Quadratic functions: Functions of the form $f(x) = ax^2 + bx + c$, where $a, b, c$ are constants.
• Exponential functions: Functions of the form $f(x) = a^x$, where $a$ is a positive constant.
• Trigonometric functions: Functions like sine, cosine, and tangent, which are based on angles.

Would you like further details on a specific type of function or an example? Here are some related questions:

1. How do you determine the domain and range of a function?
2. What is the difference between a function and a relation?
3. How can you visualize a function using a graph?
4. What is an inverse function, and how is it related to the original function?
5. How do you compose two functions to create a new function?

Tip: When working with functions, always ensure that for each input, there is only one corresponding output to maintain the definition of a function.