For each one-to-one function, a) determine f^-1(x) and b)...

Key Concepts

Graphical Representation and Reflection over y = x

Graphing a function and its inverse on the same axes demonstrates a symmetry where the graph of the inverse is the mirror image of the original function with respect to the line y = x. This property visually confirms the relationship between a function and its inverse and serves as a useful tool for verifying the correctness of the derived inverse.

One-to-One Functions

A one-to-one function is defined such that each element in the function’s domain is mapped to a unique element in its range. This property ensures that the function can be reversed, meaning an inverse function exists. One-to-one functions are the foundation for the concept of inverses because they guarantee that each output is produced by exactly one input.

Inverse Functions

An inverse function essentially reverses the roles of the inputs and outputs of the original function. Finding the inverse involves swapping the variables and solving for the new output variable, with careful consideration of the original function’s domain and range. The process confirms that applying a function and then its inverse (or vice versa) returns the initial value.

Domain and Range Considerations

When determining an inverse function, it is critical to analyze the domain and range of the original function because these sets are interchanged in the inverse. The original function’s domain becomes the range of the inverse function, and the original range becomes the inverse’s domain. This interchanging is especially significant when dealing with functions that have restricted domains.

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