Title: Default for 2.3 One-to-One and Inverse Functions
Course: HELM Workbooks 2023-2024

One-to-many rules, many-to-one and one-to-one functions

One-to-many rules

Recall that a rule for a function must produce a single output for a given input. Not all rules satisfy this criterion. For example, the rule ‘take the square root of the input’ cannot be a rule for a function because for a given input there are two outputs; an input of \(4\) produces outputs of \(2\) and \(-2\). The figure below shows two ways in which we can picture this situation, the first being a block diagram, and the second using two sets representing input and output values and the relationship between them.

[Image: Graphical representation of a how a function transforms the input to the output]
This rule cannot be a function - it is a one-to-many rule

Such a rule is described as a one-to-many rule. This means that one input produces more than one output. This is obvious from inspecting the sets in the figure above.

The graph of the rule ‘take \(\pm\sqrt{x}\)’ can be drawn by constructing a table of values:

\(x\) | 0 | 1 | 2 | 3 | 4
\(y=\pm\sqrt{x}\) | 0 | \(\pm 1\) | \(\pm \sqrt{2}\) | \(\pm\sqrt{3}\) | \(\pm 2\)

The graph is shown in the figure to the left below. For each value of \(x\) there are two corresponding values of \(y\). Plotting a graph of a one-to-many rule will result in a curve through which a vertical line can be drawn which cuts the curve more than once as you can see. The vertical line cuts the curve more than once because there is more than one \(y\) value for each \(x\) value.

[Image: a) A one-to-many 'function'; b) A one-to-one function]

By describing a rule more carefully it is possible to make sure a single output results from a single input, thereby defining a valid rule for a function. For example, the rule ‘take the positive square root of the input’ is a valid function rule because a given input produces a single output. The graph of this function is displayed to the right in the figure above.

Try for yourself: Explain why a one-to-many rule cannot be a function.

Many-to-one and one-to-one functions

Consider the function \(y(x)=x^2\). An input of \(x=3\) produces an output of \(9\). Similarly, an input of \(-3\) also produces an output of \(9\). In general, a function for which different inputs can produce the same output is called a many-to-one function. This is represented pictorially in the figure below from which it is clear why we call this a many-to-one function.

[Image: A many-to-one function]
This represents a many-to-one function

Note that whilst this is many-to-one it is still a function since any chosen input value has only one arrow emerging from it. Thus there is a single output for each input.

It is possible to decide if a function is many-to-one by examining its graph. Consider the graph of \(y=x^2\) shown below.

[Image: y=x^2]
The function \(y=x^2\) is a many-to-one function

We see that a horizontal line drawn on the graph cuts it more than once. This means that two (or more) different inputs have yielded the same output and so the function is many-to-one.

If a function is not many-to-one then it is said to be one-to-one. This means that each different input to the function yields a different output.

Consider the function \(y(x)=x^3\) which is shown in the figure below. A horizontal line drawn on this graph will intersect the curve only once. This means that each input value of \(x\) yields a different output value for \(y\).

[Image: y=x^3]
The function \(y(x)=x^3\) is a one-to-one function

Try for yourself: By sketching a graph of \(y=3x-1\) show that this is a one-to-one function.

Q: Exercise

Study the graphs shown below. Decide which, if any, are graphs of functions. For those which are, state if the function is one-to-one or many-to-one.

[Image: Three graphs of potential one-to-many, one-to-one, and many-to-one functions]

Graph (a) shows: [...]

Graph (b) shows: [...]

Graph (c) shows: [...]
A: not a function; a one-to-one function; a many-to-one function

Inverse of a function

We have seen that a function can be regarded as taking an input, \(x\), and processing it in some way to produce a single output \(f(x)\) as shown in part (a) of the figure below.

[Image: Graphical representation of a function and its inverse]
The second block reverses the process in the first

A natural question to ask is whether we can find another function that will reverse the process. In other words, can we find a function that will start with \(f(x)\) and process it to produce \(x\) again? This idea is also shown in part (b) of the figure. If we can find such a function it is called the inverse function to \(f(x)\) and is given the symbol \(f^{-1}(x)\).

Do not confuse the ‘\(-1\)’ with an index, or power. Here the superscript is used purely as the notation for the inverse function. Note that the composite function \(f^{-1}(f(x))=x\) as shown in figure below.

[Image: Graphical representation of a function and its inverse composed]
\(f^{-1}\) reverses the process in \(f\)

Example

Find the inverse function to \(f(x)=3x-8\).

Solution

The given function takes an input, \(x\) and produces an output \(3x-8\). The inverse function, \(f^{-1}\), must take an input \(3x-8\) and give an output \(x\). That is

\(\qquad f^{-1}(3x-8) = x \, .\)

If we introduce a new variable \(z=3x-8\), and transpose this for \(x\) to give

\(\qquad x=\dfrac{z+8}{3} \qquad \text{then}\qquad f^{-1}(z)=\dfrac{z+8}{3} \, .\)

So the rule for \(f^{-1}\) is add 8 to the input and divide the result by 3. Writing \(f^{-1}\) with \(x\) as its argument gives

\(\qquad f^{-1}(x)= \dfrac{x+8}{3} \, .\)

This is the inverse function.

Not all functions possess an inverse function. In fact, only one-to-one functions do so. If a function is many-to-one the process to reverse it would require many outputs from one input contradicting the definition of a function.

Try for yourself: Explain why a many-to-one function does not have an inverse function. Give an example.

Exercise

Find the inverse of the function \(f(x) = [value]\), using the fact that the inverse function must take an input \([value]\) and produce an output \(x\). So \(f^{-1}([value])=x\).

Introduce a new variable \(z\) so that \(z=[value]\) and transpose this to find \(x\). Hence write down the inverse function:

\( f^{-1}(x) = \)
Feedback: We are given the function \(f(x) =[value].\) We introduce the new variable \(z\) so that \(z=[value]\) and transpose this to find \(x.\) So \( x = \frac{[value]-z}[value].\) \(f^{-1}(z) = \frac{[value]-z}{[value]}\). With \(x\) as its argument the inverse function is \( f^{-1}(x) = [value] \). Take care the expression is in terms of the stated independent variable.

Exercise

Explain why a one-to-many rule cannot be a function.

Exercise

By sketching a graph of \(y=3x-1\) show that this is a one-to-one function.

Exercise

Explain why a many-to-one function does not have an inverse function. Give an example.

Exercise

Find the inverse of each of the following functions:

1. \(f(x)= [value] \Rightarrow f^{-1}(x) = \)
2. \(f(x)= [value] \Rightarrow f^{-1}(x) = \)
3. \(f(x)= [value] \Rightarrow f^{-1}(x) = \)
4. \(\displaystyle{f(x)= [value] \Rightarrow f^{-1}(x) = }\)
Feedback: 1. If \(f(x)= [value]\) then we can write \( x =\displaystyle \frac{y-[value]}[value]\) and \( f^{-1}(x)= \displaystyle [value].\) 2. For \(f(x)= [value]\) we take \( f^{-1}(x) = [value].\) 3. For \(f(x)= [value]\) we take \( f^{-1}(x) = \displaystyle [value].\) 4. For \(\displaystyle{f(x)= [value]}\) we take \( f^{-1}(x) =\displaystyle [value].\)

Exercise

Illustrate why \(y = x^4\) is a many-to-one function by providing suitable examples for the following:

\( x_1 = \)

\( x_2 = \)

\( y_1 = f(x_1) =\)

\( y_2 = f(x_2) =\)
Feedback: Pick \(x_1\) and \(x_2\) values which help to explain why \(y = x^4\) is many-to-one. For example, we have \([value] \neq [value]\) and yet the value of the function at these two points is [value].

Exercise

Which of the following functions are invertible?
Feedback: A function is invertible if and only if it takes each value in its range precisely once. The functions that are not invertible here are not invertible because they take some values more than once. So the invertible functions are: [value].