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In $11-16,$ determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function.$$\left\{(x, y) : y=x^{2}+2 \text { for } 0 \leq x \leq 5\right\}$$

$\{(x, y) : y=\sqrt{x-2} \text { for } 2 \leq x \leq 27\}$

Algebra

Chapter 4

RELATIONS AND FUNCTIONS

Section 8

Inverse Functions

An Introduction to Geometry

Functions

Linear Functions

Polynomials

Oregon State University

Harvey Mudd College

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Hi, everybody. This is section 4.8. Number 16 on were given the function y equals X squared plus two when x is between zero and five inclusive. So, first, we'll just to understand this is a second degree equation that's gonna be a parabola on not all Karam Bliss will have in versus unless it's restricted. So this is just going from 0 to 5. So you wanna just sketch it just to get an idea? It's an X where it's your basic parabola that's moved up to. It would look like this, right if I were toe do that. But we're only looking at the part from 0 to 5. So we're starting here and we're going, uh, so we only have this part of the problem because of the restriction. So the question is, is it going to have an inverse? So let's see what's going to happen in verse. Well, you know, we know we could reflect it in this line. Y equals X on. We can figure out what that inverse would be. Ah, few ways to go about it so we can actually do X equals life squared. Plus two, we switch the X and y to find the inverse X minus two equals lie square. And then why would be equal to the square root of X minus two? Well, looking at the problem because we have the positive part of the parabola, right? The right side. The graph here is gonna be the positive square root. It would have had to be a negative if it was the other side, but we don't have to worry about that now, so we'll have an inverse function. The question is, where is it going to be restricted? It goes from where to where? Just like there is a restriction from here Thio here because technically, it only went from zero five, so we know we had to stop it. Some point here, we'll have to stop this one, too. So the easiest thing to do is to look at the original one and just find out a little chart really quickly. If we weren't so plug in in an X Y chart. If we're gonna plug in when x zero all the way to five, what would the y value be? So we're gonna look back over here when x 00 squared plus two is two on when exes 5 25 five squared is 25 plus two is 27 so that's going to go from 2 to 27. So when we do the inverse function and we switch, our X and Y goes from to zero all the way to 27 five. So this one now will go from when X is butts Wean two and 27 inclusive.

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