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Ontario Tech acknowledges the lands and people of the Mississaugas of Scugog Island First Nation.

We are thankful to be welcome on these lands in friendship. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. These lands remain home to many Indigenous nations and peoples.

We acknowledge this land out of respect for the Indigenous nations who have cared for Turtle Island, also called North America, from before the arrival of settler peoples until this day. Most importantly, we acknowledge that the history of these lands has been tainted by poor treatment and a lack of friendship with the First Nations who call them home.

This history is something we are all affected by because we are all treaty people in Canada. We all have a shared history to reflect on, and each of us is affected by this history in different ways. Our past defines our present, but if we move forward as friends and allies, then it does not have to define our future.

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Introduction to Functions

Let’s review some background material that you will need to study functions!

Definition of a function

Before we can begin to evaluate and work with functions, it is important to understand what a function is. Functions arise when one quantity is dependent on another. A function f is a rule that assigns to each element from one set exactly one element from another set.

Definition of a function

There are 4 ways that a function can be represented:

  1. Verbally by a description in words.
  2. Numerically by a table of values.
  3. Graphically or visually using a graph.
  4. Algebraically by an equation.

Vertical Line Test

All relationships are not necessarily functions. Remember that we want each element in the first set to correspond to only one element in the second set. One method for testing to determine if a relationship is a function is the vertical line test. If a vertical line intersects a graph at more than one point, then the graph is not the graph of a function.

Example: Use the vertical line test to determine if the following graphs represent functions.

Vertical Line Test graphs

Solution: The figure on the left is an example of a function because no vertical line can intersect the graph at more than one point (assuming the same periodic behavior continues). The figure on the right is not an example of a function because there are many vertical lines that could be drawn that would intersect the graph at two points. 

Function Notation

In many cases, a function will be written in the form of an equation, such as f(x) = 3x + 6 where f(x) is read “f of x”. Another common way to write a function is in the form y = 3x +6. In this case, x and y are the variables, where x is the independent variable and y is the dependent variable. This simply means that making a change to x will result in a change to y. Both x and y are very common variables that are often used, but be aware that other variables may be used instead.