No, the inverse of a function is not always a function. For the inverse of a function to be a function, the original function must be one-to-one, meaning that each input has a unique output. In other words, for every x, there can only be one corresponding y.
If the original function is not one-to-one, there may be multiple inputs that yield the same output. In this case, the inverse would not be able to uniquely determine the original input, resulting in a contradiction to the definition of a function.
Furthermore, even if the original function is one-to-one, the inverse may not be defined for every output. For instance, consider the function f(x) = x^2. While f(x) is one-to-one for positive and negative values of x, it is not one-to-one for non-negative values of x. Consequently, the inverse of f(x) = x^2 is not a function because it is not defined for negative outputs.
Overall, the inverse of a function is only a function if the original function is one-to-one, ensuring unique input-output relationships.
Is the inverse of a function always a function? Explain.
2 answers
use simpler terms
1 answer