Explain why the inverse of 𝑓(𝑥) = (𝑥 − 2)^2 + 3 is or is not a function. If not, how would you restrict the

Question

domain of the original function 𝑓(𝑥) = (𝑥 − 2)^2 + 3 to guarantee that the inverse relation is a function?

Answer 1

The inverse of a function exists only if the function is one-to-one, meaning each input corresponds to a unique output.

In the function 𝑓(𝑥) = (𝑥 − 2)^2 + 3, the function is not one-to-one because multiple inputs can produce the same output. For example, f(0) = f(4) = 7. This means that the inverse of this function would not be a function because it wouldn't meet the one-to-one criteria.

To restrict the domain of the original function 𝑓(𝑥) = (𝑥 − 2)^2 + 3 to guarantee that the inverse relation is a function, we need to make the function one-to-one. This can be done by restricting the domain to exclude the values that create multiple outputs. In this case, we can restrict the domain to 𝑥 ≥ 2 or 𝑥 ≤ 2 to ensure that each input corresponds to a unique output. This will make the inverse of the function a function as well.