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One-to-One Functions: The Exceptional Geometry Rule

You need to understand one-to-one functions to grasp other concepts, like inverse functions. But first, let’s start with the definition of a function. A function is a geometric rule that shows a relationship between two sets of numbers. These ordered pairs of numbers are called the domain of the function (the input values) and the range of the function (the output values). In any given function, only one output value can be paired with a given input value.

See Function F below. This set of numbers is a function because no two outputs, or range values, have the same input, or domain values. However, it isn’t a one-to-one function. Let’s explore what this is.

one-to-one function: Function F diagram

The Function of One-to-One Functions

one-to-one function: Function G diagram

In one-to-one functions, no two elements, or inputs, of the domain can have the same elements, or outputs, in the range. What differentiates one-to-one from standard functions is that the inverse function is also true: No two elements of the range can have the same values within the domain.

See the picture above of Function G. This is an example of a one-to-one function because for every input there is only one output and vice versa. This concept is similar to biconditional statements, which is when a combination of two conditional statements is still true when it’s written in the reverse order:

The algebraic formula of one-to-one functions

Above is the algebraic formula of one-to-one functions. One-to-one functions are injective, meaning that only one value of x can be paired with a unique y, and only one y value can be paired with a unique x. To qualify as a one-to-one function, the following must also be true:

Condition of a one-to-one function that must be true

Proving Functions With Line Tests

Vertical line test diagram

To qualify as a function, the graph of the function must pass the vertical line test. When you draw a vertical line through each of the above graphs, you can see that each x value is only used once. Because the vertical line only intersects with one point at a time, each graph passes the vertical line test.

However, these aren’t one-to-one functions because the opposite is also true. Notice that every graph has a parabola shape. This means there are y values that coordinate with more than one x value. In other words, a horizontal line would pass through two points at once. So, at least two values of the input x have the same y, or output, value. That’s why they are not one-to-one functions. Take a look at the below examples of passing and failing line tests:

Graph showing examples of both vertical and horizontal line tests

The graph of a function lets you tell if it passes both the line tests, which would mean it’s a one-to-one function. Above, you can see graphical proof that the third example passes both tests because every value on the x-axis has a different y output. Unlike in the first two graphs, the third has no repeated x or y values.

Can You ‘Function’?

In math, one rule must often be established before another rule can be built upon it, like the Pythagorean Theorem and the converse of the Pythagorean Theorem. When you’ve established the basic principles of functions, you can understand that a one-to-one function is when there is only one element of x that can be paired with a given y and vice versa. Using vertical and horizontal line tests on a graph, you can then further determine whether a function is one-to-one.

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