In Exercises 59-62, determine whether the function is one-to-one. If it is, find its inverse fun

step 1 Okay, so here we have a modular function and we need to determine that the function is one to on

In Exercises 59-62, determine whether the function is...

Key Concepts

One-to-One Function

A one-to-one function is a function in which every element of the range corresponds to exactly one element of the domain. This means that if f(a) = f(b), then a must equal b. A useful graphical test for this property is the horizontal line test, where any horizontal line intersects the graph of the function at most once.

Inverse Function

An inverse function reverses the mapping of the original function, meaning it swaps the roles of the inputs and outputs. For a function to have an inverse, it must be one-to-one so that each output corresponds uniquely to an input. The process of finding an inverse generally involves solving the equation for x in terms of f(x) and then interchanging the variables.

Domain Restriction

Domain restriction involves limiting the set of input values for a function to ensure certain properties, such as being one-to-one. In cases where a function is not one-to-one over its entire natural domain, restricting the domain can make the function invertible by eliminating values that would otherwise produce duplicate outputs.

Absolute Value Function

The absolute value function returns the non-negative magnitude of a number. It can potentially map multiple input values to the same output, making it not one-to-one over its natural domain. However, when its domain is restricted appropriately, it can behave as a one-to-one function, allowing for the determination of an inverse function.

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