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CCSS Math: HSF.BF.B.4, HSF.BF.B.4a

We have the function f of
x is equal to x minus 1 squared minus 2. And they've constrained the
domain to x being less than or equal to 1. So we have the left half
of a parabola right here. They've constrained so that
it's not a full U parabola. And I'll let you think about
why that would make finding the inverse difficult. But let's try to find
the inverse here. And a good place to start
-- let's just set y being equal to f of x. You could say y is equal to
f of x or we could just write that y is equal to x
minus 1 squared minus 2. We know it's for x is less than or equal to 1. But right now we have y
solved for in terms of x. Or we've solved for y, to find
the inverse we're going to want to solve for x in terms of y. And we're going to
constrain y similarly. We could look at the graph and
we could say, well, in this graph right here, this is
defined for y being greater than or equal to negative 2. So we can maybe put in
parentheses y being greater than or equal to negative 2. Because this is going to be --
right now this is our range. But when we swap the x's
and y's, this is going to be our domain. So let's just keep that in
parentheses right there. So let's solve for x. So that's all you have to
do, to find the inverse. Solve for x and make sure
you keep track of the domains and the ranges. So, let's see. We could add 2 to both
sides of this equation. We get y plus 2 is equal
to x minus 1 squared. Minus 2, plus 2, so
those cancel out. And then, I'm just going to
switch to the y constraint. Because now it's not clear
what we're -- whether x is the domain or the range. But we know by the end
of this problem, y is going to be the domain. So let's just swap this here. So, 4 for y is greater than
or equal to negative 2. And we could also say, in
parentheses, x is less than 1. This is -- we haven't solved it
explicitly for either one, so we'll keep both of them
around right now. Now, to solve for x, you might
be tempted to just take the square root of both sides here. And you wouldn't be
completely wrong. But we have to be very,
very, very careful here. And this might not be something
that you've ever seen before. So this is an
interesting point here. We want the right side
to just be x minus 1. That's our goal here, in taking
the square root of both sides. We want to just have an
x minus 1 over there. Now, is x minus 1 a positive
or a negative number? Well, we've constrained our
x's to being less than 1. So we're dealing only in
a situation where x is less than or equal to 1. So if x is less than or equal
to 1, this is negative. This is negative. So we want to take the
negative square root. Let me just be very clear here. If I take negative 3 -- I take
negative 3 and I were to square it, that is equal to 9. Now, if we take the square root
of 9 -- if we take the square root -- let's say we take
the square root of both sides of this equation. And our goal is to get
back to negative 3. If we take the
positive square root. If we just take the principle
root of both sides of that, we would get 3 is equal to 3. But that's not our goal. We want to get back
to negative 3. So we want to take the negative
square root of our square. So because this expression is
negative and we want to get back to this expression, we
want to get back to this x minus 1, we need to take the
negative square root of both sides. You can always -- every
perfect square has a positive or a negative root. The principle root
is a positive root. But here we want to take the
negative root because this expression right here is
going to be negative. And that's what we
want to solve for. So let's take the negative
root of both sides. So you get the negative square
root of y plus 2 is equal to -- and I'll just write this
extra step here, just so you realize what we're doing. Is equal to the negative square
root, the negative square root of x minus 1 squared. For y is greater than or
equal to negative 2. And x is less than
or equal to 1. That's why the whole reason
we're going to take the negative square there. And then this expression
right here -- so let me just write the left again. Negative square root of y plus
2 is equal to the negative square root of x minus 1
squared is just going to be x minus 1. It's just going to
be x minus one. x minus 1 squared is
some positive quantity. The negative root is the
negative number that you have to square to get it. To get x minus 1 squared. So that just becomes x minus 1. Hopefully that doesn't
confuse you too much. We just want to get rid
of this squared sign. We want to make sure we
get the negative version. We don't want the positive
version, which would have been 1 minus x. Don't want to confuse you. So here, we now just
have to solve for x. Add 1. And let me write the 4. y is greater than or
equal to negative 2. Add 1 to both sides. You get negative square root of
y plus 2 plus 1 is equal to x for y is greater than or
equal to negative 2. Or, if we want to rewrite it,
we could say that x is equal to the negative square root of y
plus 2 plus 1 for y is greater than or equal to negative 2. Or if we want to write it in
terms, as an inverse function of y, we could say -- so we
could say that f inverse of y is equal to this, or f inverse
of y is equal to the negative square root of y plus 2 plus
1, for y is greater than or equal to negative 2. And now, if we wanted
this in terms of x. If we just want to rename
y as x we just replace the y's with x's. So we could write f
inverse of x -- I'm just renaming the y here. Is equal to the negative square
root of x plus 2 plus 1 for, I'm just renaming the y, for
x is greater than or equal to negative 2. And if we were to graph
this, let's see. If we started at x is equal
to negative 2, this is 0. So the point negative 2, 1 is
going to be on our graph. So negative 2, 1 is going
to be on our graph. Let's see, if we go to negative
1, negative 1 this will become a negative 1. Negative 1 is 0 on our graph. Negative 1, 0 is on our graph. And then, let's see. If we were to do, if we were
to put x is equal to 2 here. So x is equal to 2 is 4. 4 square root,
principle root is 2. It becomes a negative 2 So
it becomes 2, negative 1. So that's on our
graph right there. So the graph is going
to look something like this, of f inverse. It's going to look something
like that right there. As you can see, it is a
reflection of our original f of x along the line
y is equal to x. Along the line y is equal to x. Because we've essentially just
swapped the x and the y. This is about as hard of
an inverse problem that I expect you to see. Especially in a precalculus
class because it really is tricky to realize that you
have to take the negative square root here. Because the way our domain was
constrained, this value right here is going to be negative. So to solve for it, you want to
have the negative square root.