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Numerade Educator

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Problem 24 Medium Difficulty

Find a formula for the inverse of the function.

$ y = x^2 - x $ , $ x \ge \frac{1}{2} $

Answer

$\sqrt{x+\frac{1}{4}}+\frac{1}{2}=f^{-1}(x)$

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Video Transcript

Let's find the ember seven function. So we start by switching X and y, so that would give us X equals Y squared minus y. Now we went to isolate Why? So this is a little tricky because it's very difficult to isolate. Why in a quadratic function, unless you change it to a different form. So what I'm going to do is complete the square. And to do that, I'm going to figure out what I need to add to make a perfect try. No meal square. So we've y squared minus one. Why? Plus now, to get the number that we add, we take half of the coefficient on Why? So that would be negative. 1/2 and we square it, and that would be 1/4. So we need to add 1/4. Now. We can't just add 1/4 to 1 side of the equation. We have to add it to both sides of the equation. So let's add it to the other side as well. So we have X plus 1/4. All right. Now we can rewrite our perfect try. No meal square in its factored form. Why minus 1/2 quantity squared. Okay, remember, our goal is to isolate why. So let's go ahead and square root both sides of the equation now and we get the square root of X plus 1/4 equals Why minus 1/2. And finally, we're going to add 1/2 to both sides. So we have the square root of X plus 1/4 plus 1/2 equals Why? And instead of calling it why we want to call it f of X or f inverse of X. Well, actually, in this case, we don't have to, because the original function was not called F of X. So I suppose we don't have to call this f in verse, but if we want to, I think the answer key has FM for so I'll go along with that and there we have it.