3 March

A function is a simple concept that involves taking an input (often denoted as x) and producing an output (denoted as y). The relationship between the input and output is defined by the function.
Each input is related to a specific output, forming a mapping from the domain (set of possible inputs) to the range (set of possible outputs).

Functions can be represented in a table format, where one column lists inputs and the other lists the corresponding outputs.
Example: If hours worked (x) are inputs, and cups of tea consumed (y) are outputs, a table might show values like:
- 0 hours -> 0 cups
- 1 hour -> 0 cups
- 2 hours -> 0 cups
- 3 hours -> 1 cup
While useful for small domains, tables become impractical with continuous or infinite inputs.

Graphs provide a visual representation of a function, with the input (x) plotted on the horizontal axis and the output (y) on the vertical axis.
This method helps to see the overall behavior of the function and its relationship.

An equation provides a precise, mathematical representation of a function, allowing for exact outputs based on given inputs.
E.g., for the function y = f(x), if x = 2, we can directly compute y.

The domain of a function is the set of all valid inputs. For instance, inputs like negative hours worked (e.g., -1 hours) are typically not valid.
Generally, for many functions in this course, the domain will consist of subsets of real numbers.

The range is the set of all valid outputs that correspond to the inputs in the domain.
It includes y-values for which there exists at least one x-value in the domain that yields that y-value.
E.g., a function yielding negative cups of tea would not make sense in this context.

The co-domain is a broader concept of potential outputs, which may include values not actually produced by the function.
For example, while the range of a function may only include positive outputs, the co-domain could be all real numbers.

To determine if a graph represents a function, apply the vertical line test:
- A vertical line drawn across the graph should intersect the function at most once. If it intersects more than once, it indicates that the same x-value corresponds to multiple y-values, violating the definition of a function.

Cost functions relate the total cost of production to the quantity produced. E.g., if producing x items has associated costs, we denote it as C(x).
Cost functions are expected to be non-decreasing as production increases and typically include fixed and variable costs.

Revenue functions denote the total income from sales, usually described as the selling price per item times the number of items sold. E.g., R(x) = price * x.
Revenue increases linearly with respect to the number of items sold.

Profit functions are derived from subtracting costs from revenue: P(x) = R(x) - C(x).
They can yield positive (profit), negative (loss), or zero (break-even) outcomes.
The break-even point is where total revenue equals total cost, indicating no profit or loss.

A polynomial function is defined as f(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_1 * x + a_0, where a_n...a_0 are real numbers and n is a non-negative integer.
Common types include:
- Constant Function (e.g., f(x) = 5): Degree 0
- Linear Function (e.g., f(x) = 5x - 3): Degree 1
- Quadratic Function (e.g., f(x) = x^2): Degree 2
- Cubic Function (e.g., f(x) = x^3): Degree 3
Higher-degree polynomials exhibit more complex behaviors and are broadly applicable across various contexts.

Functions are versatile mathematical constructs that model relationships between inputs and outputs. Understanding their components—domain, range, representation—along with specific applications in economics, equips students with fundamental tools for analysis and problem-solving in mathematical and real-world scenarios.