Question

Explain why a function must be one-to-one in order to have an inverse that is a function. Use the function \(y=x^{2}\) to support your explanation.

Short answer

The function \( y=x^{2} \) is not one-to-one because different inputs can produce the same output. Therefore, it cannot have an inverse that is also a function, as the inverse must provide a unique output for each input.

Step-by-step solution

Understanding One-to-One Functions

A function is considered one-to-one (or injective) if and only if each element of the codomain is mapped by at most one element of the domain. In other words, if no two different inputs produce the same output.

Inverse Function Definition

For a function to have an inverse that is also a function, each element of the codomain must correspond to exactly one element of the domain. The inverse essentially reverses the mapping.

Examining the Function y=x^2

Consider the function defined by \[ y = x^2 \] If you input both 2 and -2 into this function, the output for both is 4 (i.e., \[ 2^2 = 4 \] and \[ (-2)^2 = 4 \]). This shows that different inputs can produce the same output, violating the one-to-one condition.

Impact on the Inverse

Because \[ y = x^2 \] is not one-to-one, its inverse would not assign a unique output to every input. For instance, if you tried to construct an inverse function for the input value 4, it would be unclear if the original input was 2 or -2. Thus, the inverse does not exist as a proper function.

Key concepts

One-to-One Functions

To understand why a function needs to be one-to-one to have an inverse function, let's start by defining what a one-to-one function is. A function is called one-to-one (or injective) if each element in the codomain (the set of all possible outputs) is mapped to by exactly one element in the domain (the set of all possible inputs).

This means:

• If you have two different inputs, they will produce two different outputs.
• If you find two outputs that are the same, it must mean they came from the same input.

Why is this important? It ensures no ambiguity in reversing the function, which we'll explore in the next sections.

Injective Functions

The term 'injective' is used interchangeably with 'one-to-one.' When a function is injective, it essentially means that every possible output is the result of exactly one input.

For example, consider the function defined as \( y = x^2 \). If you input both 2 and -2 into this function, the output for both is 4. This violates the injective property because different inputs yield the same output.

Therefore, \( y = x^2 \) is not injective. An injective function would ensure that if \( f(a) = f(b)\), then necessarily \( a = b \). Injective functions prevent the type of ambiguity we see with functions like \( y = x^2 \).

Function Inverses

An inverse function essentially reverses the mapping of the original function. For an inverse function to also be a function, each element in the codomain must map back to exactly one element in the domain.

Consider \( y = x^2 \) again. Since \( y = x^2 \) maps both 2 and -2 to 4, its inverse would not know whether to map 4 back to 2 or -2. This lack of uniqueness means that the inverse is not a proper function.

A proper inverse function needs the original function to be one-to-one. Only then can we ensure that the inverse mapping is clear and unambiguous.

In summary, for a function to have an inverse that is also a function, the original function must be one-to-one or injective. This ensures every input-output pair is unique and reversible.