91Ó°ÊÓ

A one-to-one function, also known as an injective function, is a fundamental concept in mathematics. It relates to how the elements of one set (typically the domain) map to unique elements of another set (the codomain).

• In a one-to-one function, each value of the domain corresponds to exactly one unique value in the codomain.
• This means no two different inputs produce the same output. Everything is distinct and non-repeating.

Think of each input-output pair as a unique match, like assigning each student in a class their own specific locker.

This property is essential because it ensures that the inverse of the function (if we can find it) will also be a function. Intuitively, knowing a function is one-to-one helps us figure out that we can reverse its process consistently.

Function inversion refers to the process of "reversing" a function. We swap the roles of inputs and outputs to reveal new relationships in our function set.

• Finding an inverse means determining an operation that "undoes" the effect of the original function.
• Notation-wise, the inverse of a function \(f\) is denoted by \(f^{-1}\).

When you apply a function and its inverse sequentially, you effectively accomplish nothing, leaving you with the initial input.

The exercise underscores this; we were given \(f^{-1}(-2) = -1\) indicating that when we plug \(-2\) into the inverse, we get \(-1\). The essence is that if we have an output \(b\) from the function, putting \(b\) back through the inverse maps it back to its original input \(a\).

Indeed, for a function \(f\) and its inverse \(f^{-1}\), they must satisfy:

• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

Thus, reversing the function returns us to our starting point, highlighting the beautiful symmetry inherent in function inversion.

Understanding the properties of functions, especially inverses, can drastically simplify problem-solving.

Key properties of inverse functions include:

• An inverse exists only for functions that are one-to-one, affirming their importance.
• The graphs of inverse functions are reflections across the line \(y = x\). This symmetric property visually represents the interchangeability of inputs and outputs.
• If \(f(a) = b\), then it directly follows that \(f^{-1}(b) = a\).

Consider the step-by-step solution about \(f^{-1}(-2) = -1\). This implies immediately that \(f(-1) = -2\), utilizing the direct relationship between a function and its inverse.

By knowing these properties, when given any task involving inverses, one can promptly solve it by using these characteristics to establish what is essentially a switch-back operation, reflecting back to original input values from their outputs.