What is the inverse of the function y = x^2 - 1?

To find the inverse of the function given by y = x² - 1, we start by swapping x and y. This gives us x = y² - 1. To solve for y, we add 1 to both sides, resulting in y² = x + 1.

Next, we need to take the square root of both sides to determine y. This provides us with the two possible solutions for y: y = √(x + 1) and y = -√(x + 1). In the context of the function's inverse, we consider both the positive and negative square roots, since the original function is not one-to-one over its entire domain.

Thus, the inverse functions can be expressed as y = ±√(x + 1). This notation indicates that for each value of x in the range of the original function, there are generally two corresponding values of y in the inverse function, namely one positive and one negative.

This comprehension of the inverse function illustrates that option C, which includes both possible values through the ± notation, accurately reflects the solutions obtained from the inverse process. The inclusion of both roots ensures that the inverse encompasses all values resulting from the original quadratic function.