Introduction to Function Inverses Introduction to Function Inverses
Introduction to Function Inverses
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- Let's think about what functions really do, and then
- we'll think about the idea of an inverse of a function.
- So let's start with a pretty straightforward function.
- Let's say f of x is equal to 2x plus 4.
- And so if I take f of 2, f of 2 is going to be equal to 2 times
- 2 plus 4, which is 4 plus 4, which is 8.
- I could take f of 3, which is 2 times 3 plus 4,
- which is equal to 10.
- 6 plus 4.
- So let's think about it in a little bit more
- of an abstract sense.
- So there's a set of things that I can input into this function.
- You might already be familiar with that notion.
- It's the domain.
- The set of all of the things that I can input into that
- function, that is the domain.
- And in that domain, 2 is sitting there, you have 3 over
- there, pretty much you could input any real number
- into this function.
- So this is going to be all real, but we're making it a
- nice contained set here just to help you visualize it.
- Now, when you apply the function, let's think about
- it means to take f of 2.
- We're inputting a number, 2, and then the function is
- outputting the number 8.
- It is mapping us from 2 to 8.
- So let's make another set here of all of the possible values
- that my function can take on.
- And we can call that the range.
- There are more formal ways to talk about this, and there's a
- much more rigorous discussion of this later on, especially in
- the linear algebra playlist, but this is all the different
- values I can take on.
- So if I take the number 2 from our domain, I input it into the
- function, we're getting mapped to the number 8.
- So let's let me draw that out.
- So we're going from 2 to the number 8 right there.
- And it's being done by the function.
- The function is doing that mapping.
- That function is mapping us from 2 to 8.
- This right here, that is equal to f of 2.
- Same idea.
- You start with 3, 3 is being mapped by the function to 10.
- It's creating an association.
- The function is mapping us from 3 to 10.
- Now, this raises an interesting question.
- Is there a way to get back from 8 to the 2, or is there a
- way to go back from the 10 to the 3?
- Or is there some other function?
- Is there some other function, we can call that the inverse
- of f, that'll take us back?
- Is there some other function that'll take
- us from 10 back to 3?
- We'll call that the inverse of f, and we'll use that as
- notation, and it'll take us back from 10 to 3.
- Is there a way to do that?
- Will that same inverse of f, will it take us back from--
- if we apply 8 to it-- will that take us back to 2?
- Now, all this seems very abstract and difficult.
- What you'll find is it's actually very easy to solve for
- this inverse of f, and I think once we solve for it, it'll
- make it clear what I'm talking about.
- That the function takes you from 2 to 8, the inverse will
- take us back from 8 to 2.
- So to think about that, let's just define-- let's just
- say y is equal to f of x.
- So y is equal to f of x, is equal to 2x plus 4.
- So I can write just y is equal to 2x plus 4, and this once
- again, this is our function.
- You give me an x, it'll give me a y.
- But we want to go the other way around.
- We want to give you a y and get an x.
- So all we have to do is solve for x in terms of y.
- So let's do that.
- If we subtract 4 from both sides of this equation-- let me
- switch colors-- if we subtract 4 from both sides of this
- equation, we get y minus 4 is equal to 2x, and then if we
- divide both sides of this equation by 2, we get y over 2
- minus 2-- 4 divided by 2 is 2-- is equal to x.
- Or if we just want to write it that way, we can just swap the
- sides, we get x is equal to 1/2y-- same thing as
- y over 2-- minus 2.
- So what we have here is a function of y that
- gives us an x, which is exactly what we wanted.
- We want a function of these values that map back to an x.
- So we can call this-- we could say that this is equal to--
- I'll do it in the same color-- this is equal to f inverse
- as a function of y.
- Or let me just write it a little bit cleaner.
- We could say f inverse as a function of y-- so we can have
- 10 or 8-- so now the range is now the domain for f inverse.
- f inverse as a function of y is equal to 1/2y minus 2.
- So all we did is we started with our original function, y
- is equal to 2x plus 4, we solved for-- over here, we've
- solved for y in terms of x-- then we just do a little bit of
- algebra, solve for x in terms of y, and we say that that is
- our inverse as a function of y.
- Which is right over here.
- And then, if we, you know, you can say this is-- you could
- replace the y with an a, a b, an x, whatever you want to do,
- so then we can just rename the y as x.
- So if you put an x into this function, you would get f
- inverse of x is equal to 1/2x minus 2.
- So all you do, you solve for x, and then you swap the y and the
- x, if you want to do it that way.
- That's the easiest way to think about it.
- And one thing I want to point out is what happens when you
- graph the function and the inverse.
- So let me just do a little quick and dirty
- graph right here.
- And then I'll do a bunch of examples of actually solving
- for inverses, but I really just wanted to give
- you the general idea.
- Function takes you from the domain to the range, the
- inverse will take you from that point back to the original
- value, if it exists.
- So if I were to graph these-- just let me draw a little
- coordinate axis right here, draw a little bit of a
- coordinate axis right there.
- This first function, 2x plus 4, its y intercept is going to be
- 1, 2, 3, 4, just like that, and then its slope will
- look like this.
- It has a slope of 2, so it will look something like-- its graph
- will look-- let me make it a little bit neater than that--
- it'll look something like that.
- That's what that function looks like.
- What does this function look like?
- What does the inverse function look like, as a function of x?
- Remember we solved for x, and then we swapped the x
- and the y, essentially.
- We could say now that y is equal to f inverse of x.
- So we have a y-intercept of negative 2, 1, 2, and
- now the slope is 1/2.
- The slope looks like this.
- Let me see if I can draw it.
- The slope looks-- or the line looks something like that.
- And what's the relationship here?
- I mean, you know, these look kind of related, it looks
- like they're reflected about something.
- It'll be a little bit more clear what they're reflected
- about if we draw the line y is equal to x.
- So the line y equals x looks like that.
- I'll do it as a dotted line.
- And you could see, you have the function and its inverse,
- they're reflected about the line y is equal to x.
- And hopefully, that makes sense here.
- Because over here, on this line, let's take
- an easy example.
- Our function, when you take 0-- so f of 0 is equal to 4.
- Our function is mapping 0 to 4.
- The inverse function, if you take f inverse of 4, f
- inverse of 4 is equal to 0.
- Or the inverse function is mapping us from 4 to 0.
- Which is exactly what we expected.
- The function takes us from the x to the y world, and then we
- swap it, we were swapping the x and the y.
- We would take the inverse.
- And that's why it's reflected around y equals x.
- So this example that I just showed you right here, function
- takes you from 0 to 4-- maybe I should do that in the function
- color-- so the function takes you from 0 to 4, that's the
- function f of 0 is 4, you see that right there, so it goes
- from 0 to 4, and then the inverse takes us
- back from 4 to 0.
- So f inverse takes us back from 4 to 0.
- You saw that right there.
- When you evaluate 4 here, 1/2 times 4 minus 2 is 0.
- The next couple of videos we'll do a bunch of examples so you
- really understand how to solve these and are able to do
- the exercises on our application for this.
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