College Algebra: Inverse of Higher-Power Function

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Okay, so let's try a couple more quick examples of finding inverses of functions where we're just given the function. Suppose you were given f(x) = x^4? Well, you could just think of that as being y equals that, flipping the roles of x and y, and so you're saying y^4 = x, and now you want to solve that. That would require you to take the 4^th root of both sides, but actually again, we're going to have this plus or minus problem because it's an even power. So therefore, this actually has no inverse.
Another way of seeing that is just to think of the graph of this. The graph of y = x^4 is very much like a parabola. It's a little bit exaggerated. So instead of being nice and smooth, it's still smooth but it has a little bit more extreme features. You could plot points and see if, in fact, this is reasonably accurate, and then you can immediately see that, in fact, this is not one-to-one. It fails the horizontal line test--no enchilada, as they say. Where do they say that, by the way? I say it, so as I say.
Let's try this one. x³ - 7. Well, first let's just see if we have any hope of having this thing have an inverse or not. Let me sketch the graph of this because it's also a good little refresher as to what that may look like. And you'll notice that now my sketches are going to become a little bit rough, because all I care about is this notion of one-to-one-ness. So, you know, what does it look like? Well, first let's graph the x³. That one's one that I hope that you're familiar with. Goes up, goes over, comes down. It's an odd function. And what does -7 do? That shifts everything down 7 units. So I just shift the whole thing down 7 units, so take the whole picture--one, two, three, four, five, six, seven. So just shift down 7 units. But you can see that the shifting doesn't change anything with respect to one-to-one-ness. This really is one-to-one. So, in fact, this should have an inverse function. How would I find it? I'd switch the roles of x and y, so I'd have x = y³ - 7 and try to solve for y. So I see y³ = x + 7 and now I can take the cube root of both sides. Now, cube roots are completely fine to take because they can be taken of positive or negative things, and you only get one answer. So, in fact, cube roots, no problem. And so, in fact, this would be f^-1(x) if this were f(x). So here's an example where you could find that. So cube roots, no problem taking. If you're taking even roots, like square roots or fourth roots, you always have to worry about plus or minus. Those functions, in fact, do not have inverses.
You know what? That's all I have to say about this, so I think I'm just going to stop right here.
Exponential and Logarithmic Functions
Finding Function Inverses
Finding the Inverse of a Function with Higher Powers Page [1 of 1]