Question

Your friend claims that every function has an inverse. Is your friend correct? Explain your reasoning.

Short answer

No, not every function has an inverse. A function has an inverse only if it is bijective (both one-to-one and onto). For example, the squaring function is not invertible as it is not one-to-one.

Step-by-step solution

Understand Bijectivity

A bijective function is a function that is both injective (or one-to-one) i.e., every element in the domain of the function maps to a unique element in the range and surjective (onto), meaning that every element in the range has a pre-image in the domain. If a function is bijective, it means it has an inverse.

Provide Examples of Non-Invertible Functions

One of the most common examples of non-invertible functions is the squaring function \( f(x) = x^2 \). If we take \( f^{-1}(x) = \sqrt{x} \), then \( f^{-1}(f(x)) = x \) only for \( x \geq 0 \) but not for \( x < 0 \). Therefore, this function is not invertible as it is not injective.

Conclusion

It is incorrect to say that all functions have inverses. Only bijective functions (which are both injective and surjective) have inverses. Non-bijective functions like squaring function are not invertible.

Key concepts

Bijective Functions

A bijective function is a special type of relationship between two sets of elements, typically known as the domain and the range. It ensures that every element in the domain is paired with exactly one unique element in the range, and every element in the range gets paired with at least one element from the domain. This unique pairing process makes the function both injective and surjective at the same time.
This dual nature is crucial because it allows the function to have an inverse, meaning you can reverse the function without losing any information. For example, consider a function that assigns each student in a class a unique student ID. This is bijective because each student ID corresponds to exactly one student, and every student has a student ID.
Bijective functions are significant in mathematics because they guarantee that each operation can be undone or reversed, providing essential insights into the structure and behavior of mathematical entities.

Injective Functions

Injective functions, also known as one-to-one functions, map every element of the domain to a unique element in the range. This means no two different domain elements can map to the same range element. The uniqueness feature of injective functions is a vital property, but it by itself does not ensure the existence of an inverse.
For example, if you have a function that assigns each person a unique phone number, this function is injective since no two people share the same phone number. However, without the surjective property, not every number may correspond to a person, meaning the function won't necessarily cover the entire range.
Injectivity is one-half of the road to bijectivity, and thus it sets the stage for creating a potential inverse. However, without being surjective, the function cannot be bijective and, therefore, not all injective functions have inverses.

Surjective Functions

Surjective functions, or onto functions, are defined by the property that every element in the range is the image of at least one element from the domain. This means the function covers the entire range and ensures that all elements in the range are mapped from the domain.
In simpler terms, consider a scenario where every seat in a theater is filled by at least one person. This situation illustrates a surjective function if we think of each person as a domain element and each seat as a range element. Although surjective functions ensure that no element in the range is left unmapped, without injectivity, multiple domain elements could map to a single range element.
The surjectivity of a function assures us of its potential to be bijective, provided that it is also injective. However, by itself, a surjective function may not possess an inverse unless it adopts injective properties.