Find the inverse of each function that is one-to-one. {(3,-1), (5,0), (0,5), (4, 2/3)}

Verified step by step guidance

1. Identify the given function as a set of ordered pairs: \((3, -1), (5, 0), (0, 5), (4, \frac{2}{3})\).
2. To find the inverse, swap the x and y values in each ordered pair.
3. The swapped pairs will be: \((-1, 3), (0, 5), (5, 0), (\frac{2}{3}, 4)\).
4. Verify that the new set of ordered pairs is also a function by checking that each x-value is unique.
5. Conclude that the inverse function is \{(-1, 3), (0, 5), (5, 0), (\frac{2}{3}, 4)\}.

Key Concepts

One-to-One Functions

A one-to-one function, or injective function, is a type of function where each output is associated with exactly one input. This means that no two different inputs produce 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 finding the inverse, it is essential to understand how to manipulate these pairs, as the inverse will consist of pairs where the x and y values are swapped, thus forming the new function.