Determining if a Function is a Function or Not.

Last friday's class we learned how to determine if a shape is a function.

Firstly, what is a function?

Function is a relation where each element of the domain (x- values) is paired with exactly 1 element of the range (y- values).

In other words, an x- value shouldn't have more than 1 y- value.

To determine if a shape is a function or not, we do 2 tests:

1. Vertical Line Test (VLT) - This test is used when graphs are given. All you have to do is draw a vertical line through the graph. If the vertical line intersects at the graph once then it is a function.

2. Horizontal Line Test (HLT) - A test use to determine if a function is one-to-one. If a horizontal line intersects a function's graph more than once, then the function is not one-to-one.

An example is the function: f(x) = x² and x² + y² = r².

First, we need to use the VLT to determine if it is a function. If the line touches at only one point then we consider it as a function. If it crosses at more than 1 point, then it is not a function.

This is a function!

This is NOT a function!

Then we use the HLT to determine if the function is a one- to- one function. If the horizontal line touches on one point, then it is a one- to- one function. If it touches more than 1 point, then it is not a one- to- one function.

This is NOT a one- to- one function!

As you can see, the line passes on more than 1 point, meaning it is not a one- to- one function.

An example of a one- to- one function is y=x. When you do both VLT and HLT, the line touches at exactly one point making it both a function and a one- to- one function.

Another example of a one- to- one function is f(x)= x³.