Find the inverse of each function that is one-to-one. {(1, -3), (2, -7), (4, -3), (5, -5)}

Verified step by step guidance

1. Identify if the function is one-to-one by checking if each x-value maps to a unique y-value.
2. Since each x-value in the set {(1, -3), (2, -7), (4, -3), (5, -5)} maps to a unique y-value, the function is one-to-one.
3. To find the inverse, swap each pair of coordinates in the function.
4. The inverse function will be the set of swapped pairs: {(-3, 1), (-7, 2), (-3, 4), (-5, 5)}.
5. Note that the inverse is not a function because the y-value -3 is repeated, violating the definition of a function.

Key Concepts

One-to-One Functions

A one-to-one function, or injective function, is a type of function where each output is produced by exactly one input. This means that no two different inputs can map to the same output. Understanding this property is crucial for finding the inverse of a function, as only one-to-one functions have inverses that are also functions.

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, the inverse function f⁻¹ takes y and returns x. To find the inverse, one typically swaps the roles of the input and output and solves for the new output, ensuring that the function remains one-to-one.

Function Notation and Ordered Pairs

Function notation and ordered pairs are fundamental in representing functions. An ordered pair (x, y) indicates that the function maps input x to output y. When working with functions, especially in finding inverses, it is important to understand how to manipulate these pairs, such as swapping them to find the inverse and ensuring that the resulting pairs still represent a valid function.