One-to-One Functions

Some functions are ‘nicer’ than others, in a way that allows us to get a new function that ‘undoes’ what the original function did.

The purpose of this lesson is to make this idea precise.

We start with a higher-level discussion of the function property, and then define one-to-one functions.

Higher-Level Discussion of the Function Property

A function is a special type of rule. To qualify as a function, each input must have exactly one corresponding output. If the function's name is $\,f\,,$ then (using function notation), the input $\,x\,$ has the corresponding unique output $\,f(x)\,.$

Here's another way to say that each input has exactly one output:

$$ \begin{gather} \overbrace{\cssId{s9}{\text{if two inputs are the same,}}}^{\cssId{s10}{\text{if }\ x = y}}\cr\cr \overbrace{\cssId{s11}{\text{then their corresponding outputs must be the same}}}^{\cssId{s12}{\text{then }\ f(x) = f(y)}}\cr\cr \cssId{s13}{\text{if } x = y,}\ \cssId{s14}{\text{then } f(x) = f(y)} \end{gather} $$

Think about this. Two is the same as two. Suppose you drop a $\,2\,$ in a box one time, and get the output $\,3\,.$ Another time, you drop a $\,2\,$ in the box, and get the output $\,5\,.$ Same input, different outputs. Not a function. When inputs are the same, the outputs must be the same. From a graphical point of view, this means that the graph of a function passes a vertical line test.

One-to-One Functions

In the previous lesson, we talked about using a function box ‘backwards’. In order to do this, we saw that each output must have exactly one corresponding input. From a graphical point of view, this means that the graph must additionally pass a horizontal line test.

Here's another way to say that each output has exactly one input:

$$ \begin{gather} \overbrace{\cssId{s28}{\text{if two outputs are the same,}}}^{\cssId{s29}{\text{if }\ f(x) = f(y)}} \cr\cr \overbrace{\cssId{s30}{\text{then their corresponding inputs must be the same}}}^{\cssId{s31}{\text{then }\ x = y}}\cr\cr \cssId{s32}{\text{if } f(x) = f(y),}\ \cssId{s33}{\text{then } x = y} \end{gather} $$

Think about this. Let $\,f\,$ be the squaring function: $\,f(x) = x^2\,.$ Observe that $\,f(2) = f(-2)\,,$ since both $\,f(2)\,$ and $\,f(-2)\,$ are the number $\,4\,.$ Thus, two outputs from the function $\,f\,$ are the same. But, $\,2\,$ isn't the same as $\,-2\,.$

For the squaring function, two outputs can be the same, but the inputs they came from can be different. As we saw, this prevents us from using the function box ‘backwards’. The squaring function doesn't have the special property needed to ‘undo’ what it did.

In order to use a function box $\,f\,$ ‘backwards’—in order to ‘undo’ what $\,f\,$ did—the following must be true:

Whenever $\,f(x) = f(y)\,,$ $\,x = y$

This is called the one-to-one property.

Equivalent Characterizations of the Function and One-to-One Properties

Every ‘if-then’ sentence has a variety of equivalent forms. You should be able to recognize the function property and the one-to-one property, no matter how they appear:

Function Property:	if $\,x=y\,$ then $\,f(x)=f(y)\,$
One-to-one Property:	if $\,f(x)=f(y)\,$ then $\,x=y\,$

Function Property:	if $\,f(x)\ne f(y)\,$ then $\,x\ne y\,$ This is the contrapositive of the prior function row: when outputs are different, the inputs must be different.
One-to-one Property:	if $\,x\ne y\,,$ then $\,f(x)\ne f(y)\,$ This is the contrapositive of the prior one-to-one row: when inputs are different, the corresponding outputs must be different.

Function Property:	$f(x)=f(y)\,,$ if $\,x=y$
One-to-one Property:	$x=y\,,$ if $\,f(x)=f(y)$

Function Property:	$x\ne y\,,$ if $\,f(x)\ne f(y)$
One-to-one Property:	$f(x)\ne f(y)\,,$ if $\,x\ne y$

Function Property:	$x=y\,$ implies $\,f(x)=f(y)$
One-to-one Property:	$f(x)=f(y)\,$ implies $\,x=y$

Function Property:	$f(x)\ne f(y)\,$ implies $\,x \ne y$
One-to-one Property:	$x \ne y\,$ implies $\,f(x)\ne f(y)$

Function Property:	$x=y \,\Rightarrow\, f(x)=f(y)$
One-to-one Property:	$f(x)=f(y) \,\Rightarrow\, x=y$

Function Property:	$f(x)\ne f(y) \, \Rightarrow\, x\ne y$
One-to-one Property:	$x\ne y \, \Rightarrow\, f(x)\ne f(y)$

Function Property:	whenever $\,x=y\,,$ $\,f(x)=f(y)$
One-to-one Property:	whenever $\,f(x)=f(y)\,,$ $\,x=y$

Function Property:	whenever $\,f(x)\ne f(y)\,,$ $\,x\ne y$
One-to-one Property:	whenever $\,x\ne y\,,$ $\,f(x)\ne f(y)$

Function Property:	$f(x)=f(y)\,,$ whenever $\,x=y$
One-to-one Property:	$x=y\,,$ whenever $\,f(x)=f(y)$

Function Property:	$x\ne y\,,$ whenever $\,f(x)\ne f(y)$
One-to-one Property:	$f(x)\ne f(y)\,,$ whenever $\,x\ne y$

Function Property:	$x=y\,$ is sufficient for $\,f(x)=f(y)$
One-to-one Property:	$f(x)=f(y)\,$ is sufficient for $\,x=y$

Function Property:	$f(x)\ne f(y)\,$ is sufficient for $\,x\ne y$
One-to-one Property:	$x\ne y\,$ is sufficient for $\,f(x)\ne f(y)$

Summary

DEFINITION one-to-one function

A function $\,f\,$ is one-to-one if and only if whenever $\,f(x) = f(y)\,,$ $\,x = y\,.$

‘One-to-one’ is often abbreviated as ‘1-1’.
A 1-1 function is a function with an additional property.

Since it's a function:	$x = y$	$\Rightarrow$	$f(x) = f(y)$
Then, it has the additional 1-1 property:	$f(x) = f(y)$	$\Rightarrow$	$x = y$
Put together, one-to-one functions satisfy:	$x = y$	$\iff$	$f(x) = f(y)$

Since a 1-1 function is a function, its graph passes a vertical line test. The additional 1-1 property says that the graph also passes a horizontal line test.
Since a 1-1 function is a function, each input has exactly one output. The additional 1-1 property says that each output has exactly one input.
For 1-1 functions, it is as if the inputs and outputs are connected with strings!

Pick up any input, follow the string to its unique corresponding output. Pick up any output, follow the string back to its unique corresponding input.

There is a one-to-one correspondence between the inputs and outputs (hence the name). An input uniquely determines an output, and an output uniquely determines an input.
One-to-one functions have inverses, which ‘undo’ what the function did. Inverses are explored in the next few lessons.