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We have the function f of x is equal to x minus

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1 squared minus 2.

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And they've constrained the domain to x being less

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than or equal to 1.

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So we have the left half of a parabola right here.

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They've constrained so that it's not a full U parabola.

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And I'll let you think about why that would make finding

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the inverse difficult.

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But let's try to find the inverse here.

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And a good place to start -- let's just set y

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being equal to f of x.

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You could say y is equal to f of x or we could just

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write that y is equal to x minus 1 squared minus 2.

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We know it's 4.

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x is less than or equal to 1.

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But right now we have y solved for in terms of x.

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Or we've solved for y, to find the inverse we're going to want

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to solve for x in terms of y.

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And we're going to constrain y similarly.

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We could look at the graph and we could say, well, in this

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graph right here, this is defined for y being greater

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than or equal to negative 2.

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So we can maybe put in parentheses y being greater

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than or equal to negative 2.

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Because this is going to be -- right now this is our range.

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But when we swap the x's and y's, this is going

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to be our domain.

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So let's just keep that in parentheses right there.

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So let's solve for x.

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So that's all you have to do, to find the inverse.

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Solve for x and make sure you keep track of the

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domains and the ranges.

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So, let's see.

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We could add 2 to both sides of this equation.

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We get y plus 2 is equal to x minus 1 squared.

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Minus 2, plus 2, so those cancel out.

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And then, I'm just going to switch to the y constraint.

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Because now it's not clear what we're -- whether x is

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the domain or the range.

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But we know by the end of this problem, y is

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going to be the domain.

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So let's just swap this here.

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So, 4 for y is greater than or equal to negative 2.

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And we could also say, in parentheses, x is less than 1.

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This is -- we haven't solved it explicitly for either one, so

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we'll keep both of them around right now.

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Now, to solve for x, you might be tempted to just take the

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square root of both sides here.

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And you wouldn't be completely wrong.

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But we have to be very, very, very careful here.

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And this might not be something that you've ever seen before.

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So this is an interesting point here.

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We want the right side to just be x minus 1.

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That's our goal here, in taking the square root of both sides.

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We want to just have an x minus 1 over there.

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Now, is x minus 1 a positive or a negative number?

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Well, we've constrained our x's to being less than 1.

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So we're dealing only in a situation where x is

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less than or equal to 1.

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So if x is less than or equal to 1, this is negative.

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This is negative.

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So we want to take the negative square root.

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Let me just be very clear here.

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If I take negative 3 -- I take negative 3 and I were to square

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it, that is equal to 9.

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Now, if we take the square root of 9 -- if we take the square

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root -- let's say we take the square root of both

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sides of this equation.

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And our goal is to get back to negative 3.

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If we take the positive square root.

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If we just take the principle root of both sides of that, we

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would get 3 is equal to 3.

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But that's not our goal.

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We want to get back to negative 3.

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So we want to take the negative square root of our square.

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So because this expression is negative and we want to get

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back to this expression, we want to get back to this x

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minus 1, we need to take the negative square root

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of both sides.

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You can always -- every perfect square has a

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positive or a negative root.

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The principle root is a positive root.

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But here we want to take the negative root because this

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expression right here is going to be negative.

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And that's what we want to solve for.

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So let's take the negative root of both sides.

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So you get the negative square root of y plus 2 is equal to

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-- and I'll just write this extra step here, just so you

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realize what we're doing.

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Is equal to the negative square root, the negative square

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root of x minus 1 squared.

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For y is greater than or equal to negative 2.

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And x is less than or equal to 1.

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That's why the whole reason we're going to take the

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negative square there.

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And then this expression right here -- so let me

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just write the left again.

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Negative square root of y plus 2 is equal to the negative

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square root of x minus 1 squared is just going

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to be x minus 1.

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It's just going to be x minus one.

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x minus 1 squared is some positive quantity.

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The negative root is the negative number that you

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have to square to get it.

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To get x minus 1 squared.

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So that just becomes x minus 1.

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Hopefully that doesn't confuse you too much.

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We just want to get rid of this squared sign.

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We want to make sure we get the negative version.

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We don't want the positive version, which would

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have been 1 minus x.

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Don't want to confuse you.

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So here, we now just have to solve for x.

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Add 1.

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And let me write the 4.

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y is greater than or equal to negative 2.

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Add 1 to both sides.

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You get negative square root of y plus 2 plus 1 is equal to x

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for y is greater than or equal to negative 2.

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Or, if we want to rewrite it, we could say that x is equal to

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the negative square root of y plus 2 plus 1 for y is greater

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than or equal to negative 2.

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Or if we want to write it in terms, as an inverse function

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of y, we could say -- so we could say that f inverse of y

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is equal to this, or f inverse of y is equal to the negative

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square root of y plus 2 plus 1, for y is greater than

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or equal to negative 2.

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And now, if we wanted this in terms of x.

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If we just want to rename y as x we just replace

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the y's with x's.

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So we could write f inverse of x -- I'm just

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renaming the y here.

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Is equal to the negative square root of x plus 2 plus 1 for,

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I'm just renaming the y, for x is greater than or

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equal to negative 2.

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And if we were to graph this, let's see.

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If we started at x is equal to negative 2, this is 0.

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So the point negative 2, 1 is going to be on our graph.

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So negative 2, 1 is going to be on our graph.

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Let's see, if we go to negative 1, negative 1 this will

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become a negative 1.

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Negative 1 is 0 on our graph.

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Negative 1, 0 is on our graph.

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And then, let's see.

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If we were to do, if we were to put x is equal to 2 here.

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So x is equal to 2 is 4.

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4 square root, principle root is 2.

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It becomes a negative 2 So it becomes 2, negative 1.

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So that's on our graph right there.

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So the graph is going to look something like

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this, of f inverse.

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It's going to look something like that right there.

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As you can see, it is a reflection of our original

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f of x along the line y is equal to x.

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Along the line y is equal to x.

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Because we've essentially just swapped the x and the y.

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This is about as hard of an inverse problem that

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I expect you to see.

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Especially in a precalculus class because it really is

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tricky to realize that you have to take the negative

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square root here.

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Because the way our domain was constrained, this value right

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here is going to be negative.

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So to solve for it, you want to have the negative square root.