One to One Functions

1. Function Review

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.

big idea

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 value.

2. 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 only 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, for example (-2, 1) and (2, 1).

3. Determining if a Function is One-to-One

In order to determine if a function is one-to-one we can use two methods:

• Graphically, where we perform both a Vertical Line Test and a Horizontal Line Test.
• Algebraically, where we use two values and b to find and manipulate the problem to show that

3a. 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.

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.

EXAMPLE

Determine if the following graph is one-to-one.



Although it passes the Vertical Line Test, this graph does not pass the Horizontal Line Test, so this graph does not represent a one-to-one function.

EXAMPLE

Determine if the following graph is one-to-one.



This function is one-to-one because it passes both the vertical and horizontal line tests.

big idea

If a graph passes both the Vertical and Horizontal Line Tests, then the graph represents a one-to-one function.

3b. 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:

step by step

1. Given a function use two values and b to find and
2. Assume
3. Manipulate the problem to show that

If we can prove that , then the function is one-to-one.

EXAMPLE

Determine if the function is one-to-one.

First, we use two values and b to find and

Next, we set them equal to each other and then manipulate the problem to see if

	Substitute expressions for and
	Subtract 4 from both sides
	Take cube-root of both sides
	Simplify
	Our solution

Since we were able to get , the function is one-to-one.

hint

By knowing the type of function, we can automatically tell (without graphing) whether it is one-to-one.

• The graph of a quadratic function is never one-to-one. You can always find a horizontal line that passes through two points.

Lines, however, offer a bit more variety.

• If a line has either a positive or negative slope, then it is a one-to-one function.
• If the linear function has zero slope (a horizontal line), then it is NOT one-to-one.
• Vertical lines violate the condition of being a function, which means that vertical lines cannot be one-to-one functions.

EXAMPLE

Without graphing, we know the following:

Equation	One-To-One Function?	Reasoning
	No	Horizontal Line
	No	Parabola
	Yes	Line with negative slope
	Yes	Line with positive slope
	No	Not a function

summary

Reviewing functions, recall that 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. One-to-One functions not only pass the Vertical Line Test, but they also must pass the Horizontal Line Test. We can determine if a function is one-to-one graphically by looking at using these tests, Passing the Vertical Line Test means that there's exactly one value for y at any given x value and passing the Horizontal Line Test means that there's exactly one value for x at any given y value. To determine if a function is one-to-one algebraically, we need to test to see if equal b if equal