This site is structured around the course I teach at the University of Arizona, Math 120R - Calculus Preparation. Before we begin, I am going to restate the course goals here as I will occasionally make reference to these when it comes to motivating certain topics and approaches:

• To prepare students to be successful in the Calculus sequence at U of A (math 122A/B, 129, and 223).
• To help students develop and refine basic algebra skills by way of an integrated review of these skills as they are needed in the course.
• To promote problem-solving and critical thinking skills through the application of algebraic concepts to common situations.
• To enhance learning and understanding of algebraic concepts through the integrated use of graphing calculators.
• To promote and utilize the "Rule of Four": all concepts are explored algebraically, numerically, graphically, and in context with applications.
• To incorporate writing into the curriculum. Through writing about mathematics you will increase your understanding of mathematical concepts.
• To help strengthen students' general academic skills.

Mathematically, when we refer to something as a function, it must satisfy the following condition:

Given a relationship between two sets, for any given input there is exactly one output.

We can denote these in a specific order, called a point or an ordered pair: \((\text{input},\text{output})\)

When we graph these on the coordinate plane, we typically consider the horizontal axis to the be the input and the vertical axis to be the output.

In this course, we primarily deal in numbers, but it should be noted that a function can be (and is) defined for notions well beyond numerical values.

Real World Examples of Functions

1. For a given traffic light, at any point in time (input) there is exactly one light illuminated (output).
2. In an investment, for a fixed time period, the interest rate (input) and the final amount in the investment (output).
3. In an investment, for a fixed interest rate, the time it is invested (input) and the final amount in the investment (output).
4. In an investment, for a target final amount, the interest rate (input) and the amount of time required to hit goal (output).
5. In vehicle safety design, the speed of a vehicle (input) and the feet required to come to a complete stop (output).
6. Given your current altitude (input), the corresponding atmospheric pressure (output).
7. For a particular employee, the hours worked (input) and their total earnings (output).

Now, a few of the items above are rearrangments of themselves (notably, the ones about investments). This is to highlight that many situations can be considered from different perspectives as functions. The key for something to make sense as a function, is that given any particular input, there is only ever one corresponding related output.

Think about each of the items listed above and really ask/answer the question for yourself: does it make sense that the description provided ensures that there is only one output for any given input?