A defining characteristic of a function is that for every element in the domain/input, there is exactly one corresponding element in the range/output. If we look at this graphically, we we will see that the graph of a function passes the Vertical Line Test, where a vertical line is drawn, and if the graph does not touch the line in more than one place we consider this a function. See the image below.
Notice how the graph does not touch each vertical line more than once. When this occurs, we say that the graph represents a function. What this means that is that each input, or x-value, of the function corresponds to exactly one output, or y-value which is the same as an f(x) value.
Introduction to One-to-One Functions
One-to-One functions are special types of functions where every value in the domain of the function corresponds to only one value in the range and each value in the range corresponds to one one value in the domain. Notice that in the graph shown above, the function is not a one-to-one function. This is because there is at least one instance where two or more x-values result in the same y-value. In order to determine if a function is one-to-one we can use two methods. The first method is graphically where we perform both a Vertical Line Test and a Horizontal Line Test.
In a Horizontal Line Test, horizontal lines are drawn on the coordinate plane and we try to determine how many times the graph of a function touches each horizontal line. If the graph only touches each horizontal line once we say that the graph passes the Horizontal Line Test.
If a graph passes both the Vertical and Horizontal Line Tests, then the graph represents a one-to-one function.
Determining if a Function is One-to-One Graphically
Given a graph of a function, we can simply draw vertical and horizontal lines on the graph to help determine if the graph represents a one-to-one function. If the graph only touches each line once then we may be safe in saying that the graph represents a one-to-one function. Look at the example shown below.
This function is one-to-one, because it passes both the vertical and horizontal line tests.
Determining if a Function is One-to-One Algebraically
Sometimes a graph of a function may be too large to draw on a coordinate plane so it can be difficult to determine if the graph represents a one-to-one function. In such cases, is it better to determine if a function is one-to-one algebraically. To determine if a function is one-to-one algebraically we do the following:
If we can prove that a = b, then we are dealing with a one-to-one function. Let's look at an example: