One-to-one functions and the horizontal line test

What is a one-to-one function?

In this section we’ll talk about how to determine whether a graph represents a one-to-one function.

If a relation is a function, then it has exactly one $y$-value for each $x$-value. If a function is one-to-one, it also has exactly one $x$-value for each $y$-value.

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The reason we care about one-to-one functions is because only a one-to-one function has an inverse. If the function is not one-to-one, then some restrictions might be needed on the domain of the function to make it invertible.

The first way we’ll look at whether or not a function is one-to-one is using the Horizontal Line Test.

One-to-one functions with the horizontal line test

Remember that we’ve already talked about the Vertical Line Test, which is a test we use to tell us whether or not a graph represents a function. Any function passing the Vertical Line Test can only have one unique output value $y$ for any single input value $x$.

In the same way that the Vertical Line Test tells us whether or not a graph is a function, the Horizontal Line Test tells us whether or not a function is one-to-one.

The graph below passes the Horizontal Line Test because a horizontal line cannot intersect it more than once.

Basically, the Horizontal Line Test says that no $y$-value corresponds to two different $x$ values. If a function passes the Horizontal Line Test, then no horizontal line will cross the graph more than once, and the graph is said to be one-to-one.

This graph doesn't pass the Horizontal Line Test because any horizontal line between $y=-2$ and $y=2$ would intersect it more than once.

All non-horizontal linear functions are one-to-one because a horizontal line drawn anywhere will only pass through once. A look at this next graph tells us that there’s no horizontal line that intersects the graph at more than one point, so the relation is a function.

On the other hand, quadratic functions are never one-to-one. A look at the next graph shows us that it’s easy to find a horizontal line that intersects the graph at more than point, thereby proving that the function is not one-to-one.

This is one reason it’s a good idea to have an idea of what function families look like. If you’re familiar with what a group of functions look like, then you can think about the graph in your head to decide if it’s a one-to-one function. For example, quadratics form a “u” shape so a horizontal line would pass through the graph twice. That means quadratic functions are never one-to-one.

One-to-one functions algebraically

Another method to check for a one-to-one function is to think that $f(a)=f(b)$ implies $a=b$ in a one-to-one function.

Say we want to know if $f(x)=\sqrt{x-2}$ is one-to-one without drawing or visualizing the graph.

Then we think if $f(a)=f(b)$ then $a=b$ and the function is one-to-one. We know that $f(a)=\sqrt{a-2}$ and $f(b)=\sqrt{b-2}$, so we could say

$f(a)=f(b)$

$\sqrt{a-2}=\sqrt{b-2}$

$\left(\sqrt{a-2}\right)^2=\left(\sqrt{b-2}\right)^2$

$a-2=b-2$

$a=b$

So $f(x)$ is a one-to-one function.

How to use the Horizontal Line Test to determine whether or not a function is one-to-one

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Showing that the function is 1-to-1

Example 

Show that the function is one-to-one by showing $f(a)=f(b)$ leads to $a=b$.

$f(x)=\frac{x-3}{x+4}$

We’ll start by replacing $x$ with $a$, and then setting that equal to whatever we get when we replace $x$ with $b$.

$\frac{a-3}{a+4}=\frac{b-3}{b+4}$

$(a-3)(b+4)=(b-3)(a+4)$

$ab+4a-3b-12=ab+4b-3a-12$

$4a+3a=4b+3b$

$7a=7b$

$a=b$

This means $f(x)$ is a one-to-one function.

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Let’s try another example.

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On the other hand, quadratic functions are never one-to-one.

Example 

Show that $g(x)$ is not one-to-one by showing that $f(a)=f(b)$ does not imply that $a=b$.

$g(x)=x^2$

All we need is one case to show that $f(a)=f(b)$ does not imply that $a=b$. That means we can choose one example where $f(a)=f(b)$ but $a \neq b$. Consider the case when $a=2$ and $b=-2$, then $a \neq b$ but

$f(-2)=(-2)^2=4$

and

$f(2)=(2)^2=4$

Therefore, $f(2)=f(-2)$ but $2 \neq -2$. Since we’ve found one case, the function is not one-to-one.

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