Which function below is the inverse of f(x) = x² – 16?

To find the inverse of the function f(x) = x² – 16, we need to follow a few steps:

1. Start by replacing f(x) with y, so we have y = x² – 16.
2. Next, we will solve this equation for x in terms of y. To do this, we first add 16 to both sides:
3. y + 16 = x².
4. Then, to solve for x, take the square root of both sides. Keep in mind that taking the square root introduces a positive and negative solution:
5. x = ±√(y + 16).

Thus, the inverse function can be expressed as:

f^-1(y) = ±√(y + 16).

But usually, we only express one branch of the inverse function. Thus, we can consider:

f^-1(y) = √(y + 16) or f^-1(y) = -√(y + 16).

In this specific case, the range of the original function (x² – 16) is [−16, ∞), so its inverse function will have a domain of [−16, ∞). Therefore, the most common expression for the inverse function is:

f^-1(y) = √(y + 16), where y ≥ -16.

In conclusion, when looking for the specific inverse function as a single branch, we identify the inverse function as f^-1(x) = √(x + 16).