Lesson 1 - Introduction to Functions

Unit 1

Chapter 1 - Introduction to Functions

Learning Goals 

• Understand and explain the meaning of the term function, and distinguish between relations and functions.
• Use function notation to represent linear and quadratic functions given their equations, tables of values, or graphs.
• Substitute into and evaluate using functions.
• Understand and explain the meanings of the terms domain and range.
• Describe the domain and range of a function, and explain the restrictions on the domain and range arising from real-world applications.
• Apply our knowledge of reverse processes to understand the process of determining inverse functions.
• Determine the numeric and graphical representations of the inverse of a function from the numeric, graphical, or algebraic representation of the function.
• Make connections between the graph of a function and the graph of its inverse.
• Understand the relationship between the domain and range of a function and the domain and range of its inverse, and determine if the inverse relation is a function.
• Determine the algebraic representation of an inverse function given the algebraic representation of a function.
• Understand the roles of the parameters a, k, d, and c in functions, and describe these roles in terms of transformations on the graph of a function.
• Apply transformations to sketch the graphs of various functions and to determine the domain and range of transformed functions.

//

Unit Introduction

• What are you doing right now? Are you on your way to school? Maybe you just walked into this room and sat down in a chair. When was the last time you walked up a flight of stairs, turned on the air conditioner, or dropped something?
• Regardless of what you are doing, we’re willing to bet you didn’t realize that everything around you can be modelled mathematically. Some of these interactions can even be represented in multiple ways.
• “Well, so what? How is this even useful?”

//

Take the Quiz

• Take the Quiz
• We're glad you asked! To demonstrate that everyday events can be modelled mathematically, we have put together a series of videos showing scenarios that can be modelled through graphs. Take the quiz to see how capable you are of modelling real-life events.

Question 1
Meet Mike, a solutions designer working in the Technical Support Department at VES. During his free time, Mike enjoys standing next to trees and texting his friends. Mike should watch where he's going.

Which graph models Mike's distance compared to time from the starting point?

ANSWER

Question 2
This sphere has no fear! It started slowly, but it was quickly on a roll.

Which graph models the ball's height in inches off the ground compared to time?

ANSWER

Question 3
We don't know what this is, but we love it. Great form, fellas! Remember, safety first, Mike.

Which graph models the distance compared to time for the race?

// ANSWER

How did you do on those questions?

• Whether your score was perfect or not so perfect, hopefully you have a better understanding of how to model real-life situations with different types of graphs.
• The applications of modelling go far beyond these simple examples, however. For example, virtual reality is an emerging and growing technology. Although the concept has been around for some time, in recent years, virtual reality has made major advances in technology. Understanding the mathematical representations of our daily activities, actions, and interactions is crucial, as such learning enables us to create virtual realities. By representing a simple action, such as walking or running, as a function, we are able to write computer programs that analyze these functions and generate responses. For example, during a run, objects need to move past us at a faster rate than they do when we walk.
• Many types of functions and mathematical concepts are used in the creation of a realistic environment. Some of these concepts are extremely complex, such as temperature variations in a room in relation to a heating/cooling source, and some are simple, such as understanding the displacement of people in relation to the ground.
• Before we can begin to model real-life situations, we first need to understand function and their key properties. The following lessons will introduce you to the necessary concepts for understanding functions.