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

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Understand that the inverse of a function swaps each input-output pair. For the given function with pairs \( (3,-1), (5,0), (0,5), (4, \frac{2}{3}) \), the inverse will have pairs where the original outputs become inputs and the original inputs become outputs.

Write the inverse pairs by switching each coordinate: \( (-1,3), (0,5), (5,0), (\frac{2}{3},4) \).

Verify that the original function is one-to-one by checking that no two different inputs have the same output, ensuring the inverse is a function.

Express the inverse function as a set of ordered pairs or as a mapping from the new inputs to outputs, based on the switched pairs.

If needed, write the inverse function notation as \( f^{-1} \) and clearly state the domain and range based on the swapped pairs.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

One-to-One Functions

A function is one-to-one if each output corresponds to exactly one input, meaning no two different inputs share the same output. This property ensures the function has an inverse because each output can be uniquely reversed to an input.

Inverse Functions

The inverse of a function reverses the roles of inputs and outputs. For a function f, its inverse f⁻¹ satisfies f(f⁻¹(x)) = x. Finding the inverse involves swapping each input-output pair, provided the function is one-to-one.

Function as a Set of Ordered Pairs

A function can be represented as a set of ordered pairs (input, output). To find the inverse, you switch each pair to (output, input). This method works well for discrete functions given explicitly by pairs.

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