Which means we were'nt 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^1(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 (0,1)

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 going to intersect it in more than one spot.

And if the original function f, is not one to one, then f^1(x) will not exist as

a function. However, if we restrict the domain of

f(x) to x>0, 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 lets

just restrict it from zero to infinity. Then, f will be one to one.

And therefore, f^-1 will exist. So, if we look at the graph of this, here

is f(x) on the restricted domain, and it's inverse will exist.

But what are we going to choose over here for f^1(x), the positive or the negative?

We're going to 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 inifinity, this has to be the range of

the inverse. And so, the output, namely y, has to be

positive. That is, f^1(x) on this restricted domain

is equal to √x-1. And let's plot this on the graph above.

This is one, and our inverse looks like this, doesn't it?

This is f^1 which is √x-1. And look at these graphs.

Aren't they symmetric about the line y=x? And that will always be true of an

function and its inverse. The graphs will always be symetric about the line y=x.

And you should graph the two lines from example number one, you'll see that the

graphs are symetric about the line y=x. And this is how we find inverse

functions. Thank you, and we'll see you next time.

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