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

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

• A. y = sqrt(x + 1)
• B. y = -sqrt(x + 1)
• C. y = ±sqrt(x + 1)
• D. y = -x^2 + 1

Answer: (C)

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.

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.