Determine whether or not the function is one to one and, if...

Key Concepts

One-to-One Function

A one-to-one function, also known as an injective function, is one where each element of the function's domain maps to a unique element in its range. This property ensures that no two distinct input values produce the same output, which is a necessary condition for the existence of an inverse function.

Inverse Function

An inverse function reverses the effect of the original function by mapping each output back to the unique input that produced it. For a function to have an inverse, it must be one-to-one; the process of finding an inverse involves interchanging the roles of the dependent and independent variables and solving for the new dependent variable.

Domain and Range

The domain of a function consists of all possible input values, while the range consists of all outputs produced by those inputs. When a function has an inverse, the domain of the original function becomes the range of the inverse, and the range of the original function becomes the domain of the inverse.

Algebraic Manipulation for Finding an Inverse

Finding the inverse of a function typically involves expressing the function in the form y = f(x), then swapping the variables to obtain x = f(y). The next step is to solve this new equation for y, which yields the inverse function. This technique is a fundamental method in algebra for reversing the roles of input and output.

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