Graph the inverse of each one-to-one function.

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each output value is paired with exactly one input value. This means that no two different inputs produce the same output, which is crucial for the existence of an inverse function. To determine if a function is one-to-one, the horizontal line test can be applied: if any horizontal line intersects the graph more than once, the function is not one-to-one.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, then its inverse function f⁻¹ takes y as input and returns x. For a function to have an inverse, it must be one-to-one, ensuring that each output corresponds to a unique input, allowing for a clear reversal of the mapping.

Graphing Inverses

When graphing the inverse of a function, the graph of the inverse can be obtained by reflecting the original graph across the line y = x. This reflection illustrates how the roles of the input and output are switched. Understanding this geometric relationship helps in visualizing and confirming the properties of inverse functions, particularly in ensuring that the inverse is also a function.