Question

Explain what is meant by the inverse of a function.

Short answer

Answer: The inverse of a function is another function that essentially reverses or undoes the actions of the original function. It is denoted as f^{-1}. For a function to have an inverse, it must be both injective (one-to-one) and surjective (onto), making it a bijective function.

Step-by-step solution

Definition of the Inverse of a Function

In mathematics, the inverse of a function is another function that essentially "reverses" or "undoes" the actions of the original function. In a more formal definition, for a function f, its inverse function, denoted as f^{-1}, is a function such that for every input-output pair (x, y) in f, there is a corresponding input-output pair (y, x) in f^{-1}. In other words, if f(x)=y, then f^{-1}(y)=x.

Example of an Inverse Function

Let's consider the function f(x) = 2x + 3. To find the inverse of this function, we first need to solve for x in terms of y. So, let y = 2x + 3. Solving for x, we get x = (y - 3)/2. Therefore, the inverse function f^{-1}(y) = (y - 3)/2.

Graphical Representation

If you were to graph both a function and its inverse on the same set of axes, you'd find that the two graphs are reflections of each other across the line y=x. This is because the x and y-values have switched places between the function and its inverse.

Conditions for a Function to Have an Inverse

For a function to have an inverse, it must be both injective (one-to-one) and surjective (onto). Injective means that every input has a unique output, while surjective means that every output has a corresponding input. If a function is both injective and surjective, it is called a bijective function. Bijective functions have unique inverses.

The Relationship Between a Function and Its Inverse

A function and its inverse have a special property: when the inverse function is applied to the output of the original function, the original input is obtained. Mathematically speaking, this means that if f(x)=y, then f^{-1}(f(x)) = f^{-1}(y) = x. Additionally, if f^{-1}(y) = x, then f(f^{-1}(y)) = f(x) = y. These properties hold true for all valid inputs and outputs in the domain and range of the function and its inverse.

Notation for the Inverse of a Function

The notation for the inverse of a function is typically written as f^{-1}(x) (note that this does NOT mean 1/f(x)). Keep in mind that this is different from the reciprocal of a function, which would be written as 1/f(x) or f(x)^{-1}.

Key concepts

Bijective Function

A bijective function is crucial for a function to have an inverse. This type of function is both injective and surjective. Let's break this down:

• Injective (One-to-One): Each element in the domain maps to a unique element in the range. This means no two different inputs will give the same output.


• Surjective (Onto): Every element in the range is covered, meaning each output has a pre-image in the domain.

A function that covers these two properties is called "bijective." Only bijective functions can have a proper inverse. Without these properties, the function cannot "reverse" its operation accurately.

Graphical Representation of Functions

Graphing a function and its inverse provides a visual understanding of their relationship. When both are plotted on the same graph, they appear as mirror images across the line \(y = x\). This is because the original function \(f(x) = y\) has its x and y values swapped in its inverse \(f^{-1}(x) = y\).

Imagine folding a piece of paper along the line \(y=x\) and observing how the points from the graph of the function align with points on the graph of the inverse. This reflection property is significant when checking if you have correctly found an inverse.

Properties of Inverse Functions

Inverse functions have specific characteristics that highlight their unique relationship with the original function. Some important properties include:

• Cancellation Property: Applying the inverse function to the function result gives back the original input. Mathematically, this is described by \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(y)) = y\).


• Domain and Range Swap: The domain of the original function becomes the range of its inverse and vice versa. If the domain of \(f\) is \([a, b]\), the range of \(f^{-1}\) will be \([a, b]\).

These properties ensure that using inverses maintains the integrity of the original function's input-output pairs.

Function Notation

Function notation is a way to clearly define functions and their inverses. The notation for an inverse function is \(f^{-1}(x)\). It is important to understand that \(f^{-1}(x)\) does not imply the reciprocal of \(f(x)\), but rather a new function entirely.

In mathematical notation:

• \(f(x)\) denotes the original function, representing the relationship between input \(x\) and output \(y\).


• \(f^{-1}(x)\) denotes the inverse, signifying the process to retrieve the input from a given output.

This notation helps in avoiding confusion and clearly specifying operations between functions and their inverses.