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One-to-One Functions

In this lesson, we will understand the concept of one-to-one functions, which is also known as injective functions. Only one-to-one functions can have an inverse found algebraically.

Let $A$ and $B$ be sets such that there exists a function $f$ such that $f: A \rightarrow B$. The function $f$ is injective (one-to-one) if for all elements $a_1, a_2$ in domain $A$, $f(a_1) = f(a_2)$ implies that $a_1 = a_2$.

In other words, every element in the domain is mapped to exactly one element in the codomain - no more, no less! In addition, no two elements in the domain can be mapped to the same element in the codomain.

We can call an injective function an injection.

Let's take a look at an example of an injective function. Let's pretend the function $f$ maps dog owners to their dogs. The number of elements in the input group is three unique elements and in the output group is also three.

The function $f$ is injective because every owner has a dog, and no two owners have the same dog.

Now you try!

Take a look at these three different versions of function $f$.

Notice anything?

Hint: take a look again at the number of elements in the domain $A$ and the codomain $B$.

We can refer to the number of elements in a set as its size. If $f: A \rightarrow B$ is an injective function, then the size of $A$ must always be less than the size of $B$.

Why is this true?

Keeping in mind our analogy above, every dog owner must have exactly one dog, right? Then, at the very least, the size of $B$ will be equal to the size of $A$. However, for a function to be injective we only require that every element in the domain $A$ is mapped to an element in $B$. That being said, there can be elements in $B$ that are not mapped to. In other words, it is possible to have dogs without owners.

You will later learn about surjective functions, which guarantee that every dog has an owner! A function can also be called bijective if it is **both **surjective and injective.

Injection Review: Test your knowledge!

Even More About Injections: Inverses

If we know that a function is injective, we can find its inverse.

If function $f$ lets us find dogs by inputting owners, then we can find an inverse function $g$ such that we can find owners by inputting dogs. In simpler terms, we can find a function that will map dogs to their owners.

Remember that the inverse of a function is a reflection over the line $y=x$.

Let $A$ and $B$ be two sets. A function (also called map) $f:A\rightarrow B$ is said to be injective, or one-one, if no two elements of $A$ are mapped to the same element of $B$ under $f$. Mathematically, $x_1, x_2 \in A$ and $f(x_1)=f(x_2)$ $\implies$ $x_1=x_2$.

(A function is called many-one if it is not injective.)

The above arrow-diagram shows a map $f$ that is injective.

The above arrow-diagram shows a map f that is not injective, i.e., it is many-one, because $x$ and $y$ are distinct elements of A, but $f(x)=f(y)$.

(Video) Injective Functions

by learnifyable

Follow along with the examples in this video from learnifyable.

Functions can be classified as one-to-one or onto based on how their outputs relate to the range.

One-to-One Functions

A function is like a machine that takes an input and gives an output. A one-to-one functions is a special kind of function where each input has a unique output. Each output has a unique input. This means that no two different inputs can give the same output.

To check if a function is one-to-one, imagine you are looking at the graph. If a horizontal line crosses the graph more than once, then the function is not one-to-one. This is called the horizontal line test.

For example, the function $f(x)=x^2$ is not one-to-one because both $x=-2$ and $x=2$ give the same output, $4$. On the other hand, the function $f(x)=2x+3$ is one-to-one because each input results in a different output.

A quick way to see this with numbers is to pick two different inputs, like 1 and 2. If $f(1)$ and $f(2)$ are not the same, the function is one-to-one.

One-to-one functions are important in math because they help us to solve equations and make sure that each output has only one input. You can also use this property to find the inverse of the function, which "undoes" the function's work.

In summary, a one-to-one function is a function where no two different inputs share the same output. If you see a graph, check if any horizontal lines cross the graph more than once. If they do, it's not one-to-one!