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

So we have f of x is equal to
negative x plus 4, and f of x is graphed right here on
our coordinate plane. Let's try to figure out
what the inverse of f is. And to figure out the inverse,
what I like to do is I set y, I set the variable y, equal to f
of x, or we could write that y is equal to negative x plus 4. Right now, we've solved
for y in terms of x. To solve for the inverse,
we do the opposite. We solve for x in terms of y. So let's subtract 4
from both sides. You get y minus 4 is
equal to negative x. And then to solve for x, we can
multiply both sides of this equation times negative 1. And so you get negative
y plus 4 is equal to x. Or just because we're always
used to writing the dependent variable on the left-hand side,
we could rewrite this as x is equal to negative y plus 4. Or another way to write it
is we could say that f inverse of y is equal
to negative y plus 4. So this is the inverse function
right here, and we've written it as a function of y, but we
can just rename the y as x so it's a function of x. So let's do that. So if we just rename this y as
x, we get f inverse of x is equal to the negative x plus 4. These two functions
are identical. Here, we just used y as the
independent variable, or as the input variable. Here we just use x, but they
are identical functions. Now, just out of interest,
let's graph the inverse function and see how it
might relate to this one right over here. So if you look at it,
it actually looks fairly identical. It's a negative x plus 4. It's the exact same function. So let's see, if we have-- the
y-intercept is 4, it's going to be the exact same thing. The function is
its own inverse. So if we were to graph it,
we would put it right on top of this. And so, there's a couple of
ways to think about it. In the first inverse function
video, I talked about how a function and their inverse--
they are the reflection over the line y equals x. So where's the line
y equals x here? Well, line y equals
x looks like this. And negative x plus 4 is
actually perpendicular to y is equal to x, so when you reflect
it, you're just kind of flipping it over, but it's
going to be the same line. It is its own reflection. Now, let's make sure that
that actually makes sense. When we're dealing with the
standard function right there, if you input a 2,
it gets mapped to a 2. If you input a 4, it
gets mapped to 0. What happens if you
go the other way? If you input a 2, well, 2
gets mapped to 2 either way, so that makes sense. For the regular function,
4 gets mapped to 0. For the inverse function,
0 gets mapped to 4. So it actually makes
complete sense. Let's think about
it another way. For the regular function-- let
me write it explicitly down. This might be obvious to
you, but just in case it's not, it might be helpful. Let's pick f of 5. f of 5 is equal to negative 1. Or we could say, the function f
maps us from 5 to negative 1. Now, what does f inverse do? What's f inverse of negative 1? f inverse of negative 1 is 5. Or we could say that f maps
us from negative 1 to 5. So once again, if you think
about kind of the sets, they're our domains and our ranges. So let's say that this is
the domain of f, this is the range of f. f will take us from
to negative 1. That's what the
function f does. And we see that f inverse takes
us back from negative 1 to 5. f inverse takes us back
from negative 1 to 5, just like it's supposed to do. Let's do one more of these. So here I have g of x is equal
to negative 2x minus 1. So just like the last problem,
I like to set y equal to this. So we say y is equal to g
of x, which is equal to negative 2x minus 1. Now we just solve for x. y plus 1 is equal
to negative 2x. Just added 1 to both sides. Now we can divide both sides of
this equation by negative 2, and so you get negative y over
2 minus 1/2 is equal to x, or we could write x is equal to
negative y over 2 minus 1/2, or we could write f inverse as a
function of y is equal to negative y over 2 minus 1/2, or
we can just rename y as x. And we could say that f inverse
of-- oh, let me careful here. That shouldn't be an f. The original function was
g , so let me be clear. That is g inverse of y is equal
to negative y over 2 minus 1/2 because we started with a
g of x, not an f of x. Make sure we get our
notation right. Or we could just rename the y
and say g inverse of x is equal to negative x over 2 minus 1/2. Now, let's graph it. Its y-intercept
is negative 1/2. It's right over there. And it has a slope
of negative 1/2. Let's see, if we start at
negative 1/2, if we move over to 1 in the positive direction,
it will go down half. If we move over 1 again, it
will go down half again. If we move back-- so
it'll go like that. So the line, I'll try my
best to draw it, will look something like that. It'll just keep going, so it'll
look something like that, and it'll keep going in
both directions. And now let's see if this
really is a reflection over y equals x. y equals x looks
like that, and you can see they are a reflection. If you reflect this guy, if you
reflect this blue line, it becomes this orange line. But the general idea, you
literally just-- a function is originally expressed, is
solved for y in terms of x. You just do some algebra. Solve for x in terms of y, and
that's essentially your inverse function as a function of y,
but then you can rename it as a function of x.