One-to-One Functions and Inverses

Hello!

In our last lesson, we saw how composing two functions can create a new one. We even saw a special case where composing functions like $f (x) = x^{2} + 1$ and $g (x) = x - 1$ (for $x \geq 0$ ) resulted in simply $x$ . This suggests that one function "undoes" the other. This "undoing" relationship is the central idea behind inverse functions.

Today's lesson addresses the learning outcome: Determine whether a function is one-to-one and find the inverse of a function. We'll first explore the necessary condition a function must meet to have an inverse—it must be "one-to-one." Then, we will learn a reliable method to find the inverse function and investigate the relationship between a function and its inverse.

1. One-to-One Functions: The Prerequisite for an Inverse

Imagine a function as a process that takes an input and produces an output. An inverse function should reverse that process, taking the output and returning the original input.

But what if two different inputs lead to the same output? For example, with $f (x) = x^{2}$ , both $f (2) = 4$ and $f (- 2) = 4$ . If we try to reverse this, what is the inverse of 4? Is it 2 or -2? An inverse function can't produce two different answers for the same input.

This leads us to a crucial requirement: for a function to have a well-defined inverse, every output must be linked to exactly one unique input. We call such functions one-to-one.

To get a solid intuition for this, let's watch the beginning of a video from Professor Leonard. He uses a great analogy involving employee ID numbers.

One to One Functions (Precalculus - College Algebra 50)

This video, 'One to One Functions', explains the core concept of a one-to-one function and why it's a necessary condition for inverses to exist.

Watch the first 5 minutes and 22 seconds (00:00 - 05:22). Focus on the definition of a one-to-one function (one input gives one unique output) and the employee ID example, which clearly illustrates why a non-one-to-one relationship is not 'mappable backwards'.

As the video explains, a function is one-to-one if each output is unique. This means no two different inputs produce the same output. In British A-Level terminology, a one-to-one function is also called an injective function.

How to Check if a Function is One-to-One

We have two primary methods for testing if a function is one-to-one.

A. The Graphical Method: The Horizontal Line Test

You'll recall the Vertical Line Test, which checks if a graph represents a function. A similar test exists for checking if a function is one-to-one.

The Horizontal Line Test states that if you can draw a horizontal line that intersects a function's graph more than once, the function is not one-to-one. If no horizontal line intersects the graph more than once, it is one-to-one.

Horizontal Line Test for One-to-One Functions — Caption: The Horizontal Line Test applied to two functions. The linear function $f(x)=x$ is one-to-one because any horizontal line hits the graph only once. The quadratic function $f(x)=x^2$ is not one-to-one because a horizontal line can intersect it twice, meaning two different inputs produce the same output.

B. The Algebraic Method

The algebraic method directly tests the definition. We assume two different inputs, $x_{1}$ and $x_{2}$ , produce the same output and see what that implies.

Set $f (x_{1}) = f (x_{2})$ .
Solve this equation for $x_{1}$ and $x_{2}$ .
If the only possible solution is $x_{1} = x_{2}$ , the function is one-to-one. If you find another solution (e.g., $x_{1} = - x_{2}$ ), it is not one-to-one.

Example: Is $f (x) = 3 x - 5$ one-to-one?

Assume $f (x_{1}) = f (x_{2})$ .
$3 x_{1} - 5 = 3 x_{2} - 5$
$3 x_{1} = 3 x_{2}$
$x_{1} = x_{2}$
The only solution is $x_{1} = x_{2}$ , so the function is one-to-one.

Example: Is $f (x) = x^{2}$ one-to-one?

Assume $f (x_{1}) = f (x_{2})$ .
$x_{1}^{2} = x_{2}^{2}$
$x_{1}^{2} = x_{2}^{2}$
$x_{1} = \pm x_{2}$
This gives two possibilities: $x_{1} = x_{2}$ OR $x_{1} = - x_{2}$ . Because there's more than one possibility, the function is not one-to-one.

The following resource provides a good summary of these tests and uses terminology you might be familiar with.

One to one Function | Definition, Graph & Examples | A Level

This article from A-Level Maths, 'One to one Function', defines the concept and provides both the horizontal line test and the algebraic method for identification.

Read the sections 'The idea of one-one function' and 'How to identify one-one function?'. This will reinforce the definition and the two tests we just discussed.

Restricting the Domain

What if we have a function like $f (x) = x^{2}$ that isn't one-to-one, but we still want to find an inverse for it? We can often do this by restricting the domain of the original function to a section where it is one-to-one.

For $f (x) = x^{2}$ , the "U" shape fails the horizontal line test. But if we only consider the right half of the parabola (where $x \geq 0$ ), that piece of the graph does pass the test. On this restricted domain, the function is now one-to-one and we can define an inverse.

This is a common and important technique, particularly with trigonometric functions, which we will see later in the course.

Inverse functions

The document 'Inverse functions' from mathcentre.ac.uk has an excellent section explaining why and how we restrict domains.

Please read Section 4, 'Restricting domains' (pages 6-8). It uses the examples of f(x) = x² and f(x) = sin(x) to show how restricting the domain makes it possible for a function to have an inverse.

2. Finding and Understanding Inverse Functions

Now that we can identify a one-to-one function, let's find its inverse. The inverse function is denoted $f^{- 1} (x)$ .

Important: The notation $f^{- 1} (x)$ means "the inverse of f(x)". It does not mean $\frac{1}{f ( x )}$ . This is a common point of confusion. For example, if $f (x) = x + 2$ , its inverse is $f^{- 1} (x) = x - 2$ , but its reciprocal is $\frac{1}{x + 2}$ .

The Algebraic Method for Finding an Inverse

There is a straightforward, four-step procedure for finding the inverse of a function $f (x)$ .

Replace $f (x)$ with $y$ . This makes the algebra easier to handle.
Swap all instances of $x$ and $y$ . This is the crucial step that performs the "inversion."
Make $y$ the subject of the new equation (solve for $y$ ).
Replace $y$ with the inverse notation $f^{- 1} (x)$ .

The following video walks through this method with several clear examples. It's from a GCSE Higher Maths channel, so the pace and style should be very accessible.

Inverse Functions - GCSE Higher Maths

This video, 'Inverse Functions - GCSE Higher Maths', demonstrates the four-step algebraic method for finding an inverse function.

Watch from 02:05 to 06:38. This covers three examples applying the four-step method to a linear function, a fractional function, and a quadratic function, which involves a square root.

Test your understanding!

Let's find the inverse of the one-to-one function $f (x) = \frac{x + 5}{2}$ .

Follow the four steps:

Replace $f (x)$ with $y$ .
Swap $x$ and $y$ .
Make $y$ the subject.
Replace $y$ with $f^{- 1} (x)$ .

Show answer

Replace $f (x)$ with $y$ :
$y = \frac{x + 5}{2}$
Swap $x$ and $y$ :
$x = \frac{y + 5}{2}$
Make $y$ the subject:
$2 x = y + 5$
$2 x - 5 = y$
Replace $y$ with $f^{- 1} (x)$ :
$f^{- 1} (x) = 2 x - 5$

So, the inverse of $f (x) = \frac{x + 5}{2}$ is $f^{- 1} (x) = 2 x - 5$ . You can check this: if you put $x = 3$ into $f (x)$ , you get $f (3) = \frac{3 + 5}{2} = 4$ . If you then put that output, 4, into the inverse, you get $f^{- 1} (4) = 2 (4) - 5 = 3$ , which is your original input. The inverse function worked!

3. Properties of Inverse Functions

Inverse functions have two very elegant properties related to their domain, range, and graph.

Domain and Range

When you swap $x$ and $y$ to find the inverse, you are also swapping the roles of the domain and range.

The domain of $f (x)$ becomes the range of $f^{- 1} (x)$ .
The range of $f (x)$ becomes the domain of $f^{- 1} (x)$ .

This is extremely useful. If you know the domain and range of a function, you immediately know the domain and range of its inverse.

For example, consider $f (x) = x$ .

Domain of $f$ : $[0, \infty)$
Range of $f$ : $[0, \infty)$
The inverse is $f^{- 1} (x) = x^{2}$ , but its domain is restricted to the range of the original function.
Domain of $f^{- 1}$ : $[0, \infty)$
Range of $f^{- 1}$ : $[0, \infty)$

The Graph of an Inverse Function

The act of swapping $x$ and $y$ coordinates has a beautiful graphical effect: the graph of $f^{- 1} (x)$ is a reflection of the graph of $f (x)$ across the line $y = x$ .

Graph of a One-to-One Function and its Inverse — Caption: The graph of the function $g(x) = x^3$ (in blue) and its inverse $g^{-1}(x) = \sqrt[3]{x}$ (in orange). You can see that they are perfect mirror images of each other, reflected across the dashed line $y=x$.

This graphical property provides a great way to visualise an inverse and to check your work. If you sketch a function and its inverse and they don't look like reflections across $y = x$ , something has likely gone wrong.

Conclusion

This lesson connected several important ideas. We established that for a function to be reversible, it must be one-to-one, meaning each output is unique.

Key Takeaways:

A function is one-to-one if every output corresponds to exactly one input.
We can test for this property using the Horizontal Line Test on a graph or by algebraically showing that $f (x_{1}) = f (x_{2})$ implies $x_{1} = x_{2}$ .
Only one-to-one functions have inverse functions, denoted $f^{- 1} (x)$ .
We can find the inverse using a four-step algebraic method: replace, swap, solve, replace.
The graph of an inverse function is a reflection of the original function's graph across the line $y = x$ .
The domain and range of a function and its inverse are swapped.

Preview of the Next Lesson:
We will continue our analysis of function properties by looking at symmetry. We will learn how to determine if a function is "even," "odd," or neither, which describes its symmetry with respect to the y-axis or the origin. This provides another powerful tool for understanding and sketching graphs.

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