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[sound] Let’s look at finding the inverse of a function.
[sound]. For example, let f(x) equals one minus 4x,
Let’s find f inverse(x) if it exists. Remember that the inverse of a function,
is defined as follows: y is equal to f inverse (x),
if and only if, f(y) is equal to x. And remember here, be very careful with
this notation. This does not mean one divided by f.
It's just a notation for the inverse function.
So notice what we're doing here is we're switching the roles of x and y, and then
solving for y, and that will give us f inverse (x).
That is, let's let y equal f(x), or one minus 4x, and now we'll interchange the
roles of x and y, so wherever we see a y we'll put an x, and wherever we see an x
we'll put a y. And then we'll solve for y.
So we have four y, is equal to one minus x.
And then dividing both sides by four, gives us that y is equal to one minus x
divided by four. Or dividing both terms in the numerator by
Four, we get y is equal two, one-fourth minus x over four.
And we were uniquely able to solve for y, so this is an inverse.
So f inverse (x) is equal to One-fourth minus x divided by four.
Alright let’s look at another example. [sound].
Let f(x) equal x squared plus one. Let’s find f inverse (x) if it exists.
Again, we'll let y equal f(x) where y is equal to
X squared plus one, and then we'll interchange the roles of x and y, which
gives us y is equal to, y squared plus one, and then we'll solve for y.
We have y squared is equal to x minus one. Or, y is equal to plus or minus the square
root of x minus one, which means we weren't uniquely able to solve for y.
Because for any input x, we'd get two possible y values, the positive, and the
negative square root, which means that f inverse (x) does not
exist, which shouldn't surprise you if you think
of the graph of this function. It's a parabola, with vertex at zero, one,
opening upward, which isn't a one to one function, is it?
Because by the horizontal line test, if we pass a horizontal line through this graph,
it's gonna intersect it in more than one spot.
And if the original function f is not one to one,
then f inverse (x) will not exist as a function.
However, if we restrict the domain of f(x) to x greater than zero,
Then we're only looking at this part of the parabola.
Therefore, on that restricted domain, f is one-to-one.
So f, on the restricted domain, will have an inverse.
That is, if we restrict The domain of f
to the interval zero to infinity, for example,
we could have also restricted it from negative infinity to zero, but let's just
restrict it from zero to infinity. Then,
F will be one to one. And therefore,
F inverse will exist. So if we look at the graph of this.
There is f(x) on the restricted domain. Then its inverse will exist.
But what are we gonna choose over here for f inverse (x), the positive or the
negative? We're gonna choose the positive because
remember that the range of the inverse is equal to the domain of
the original function. And if we're restricting that domain,
zero to infinity, this has to be the range of the inverse.
And so the output, namely y, has to be positive.
That is, f inverse (x) on this restricted domain is equal to the square root of x
minus one. And let's plot this on the graph above.
This is one. Then our invers, looks like this, doesn't
it? This is f inverse,
square root of x minus one. Now look at these graphs. Aren't they
symmetric about the line y equal x? And that will always be true about a
function and its inverse. The graphs will always be symmetric about the line y equal
x. And you should graph the two lines from
example number one, you'll see that their graphs are symmetric about the line y
equal x. And this is how we find inverse functions.
Thank you, and we'll see you next time. [sound]