A **one to one function** is a relation that preserves “distinctness”; Every unique member of the domain is mapped to a unique member of its range. Sometimes this type of mapping is called *injective*.

Both images below show injective functions. Notice that each element in X is mapped to a distinct element of Y.

The image below is *not* this type of mapping. Two distinct elements of X (3 and 4), are mapped to the same element of Y (C).

## The Horizontal Line Test for a One to One Function

- One easy way of determining whether or not a mapping is injective is the horizontal line test.

Graph the function. - Draw a horizontal line over that graph.
- If any horizontal line intersects the graph of the function more than once, the function is not one to one.

But if there *isn’t* any straight horizontal line that can be drawn to cross the function more than once, the function *is* one to one.

Below are images of two different mappings. On the first graph, you could draw a horizontal line that cuts through the hump and crosses the mapping twice. On the second graph, though, there is no way we can draw a horizontal straight line that crosses twice. The function in this second graph is therefore one to one.

## Properties of One to One Functions

Here are some useful properties of injective functions:

- If two functions
*f*and*g*are both one to one, so is*f*·*g*. - If
*g*·*f*is one to one,*f*is as well. - If
*f*is injective, it has an inverse function (a function that undoes it).

## References

He, Jiwen. Inverses and More. Lecture 1, Section 7.1 Notes. Retrieved from https://www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture01_handout.pdf

Oldham, K. (2008). An Atlas of Functions. Springer.

University of Toronto Math. Preparing for Calculus: Functions and Their Inverses. Retrieved from https://www.math.toronto.edu/preparing-for-calculus/4_functions/we_3_one_to_one.html on June 15, 2019.