Question

How do I find the inverse of f(x)=x^2-16,x>=0

like, what do I do with the x>=0?

Answer 1

The x >= to 0 is required because without it, x^2 - 16 does not have an inverse by definition ( it would not pass the vertical line test).

In order to actually get the inverse, we can look at f(x) = x^2 - 16 as y = x^2 - 16. Now we're required to switch x and y, giving us x = y^2 - 16. From here, we solve for y

\(x = y^{2} - 16\)
\(x + 16 = y^{2}\)
\(\pm\sqrt{x+16} = y\)

If we included the plus or minus as a part of the answer, we would not have an actual inverse, so we need to pick one, the plus or the minus. Which one we pick is determined by how x was defined. It gave us that x had to be greater than 0. This means we want the positive answer, so our inverse would be \(y = \sqrt{x + 16}\)

Because of how inverse are graphed, you can actually always check to see if you chose the right sign, plus or minus, of your inverse. Know that a function and its inverse are symmetric about the line y = x. So if we were to graph our function and our solution, we would have this:



If I would have chose \(-\sqrt{x+16}\) I would not have this symmetry. Had the problem told me x <= 0, I wouldve actually needed to choose the negative answer. Hope this makes sense : )