Inverse function - Wikipedia
Click on the gift to see the function and the inverse of the function. Describe the difference between the first table and the second table. Interactive popup. Examine and use the properties of the inverse functions. Putting “Inverse-Square Law” in Conceptual terms: The Inverse-Square Law is a the function is: y=k/x2 and y is said to be inversely proportional to the square of x. The relationship will be a curve that approaches the x and y axes but will.
This work can sometimes be messy making it easy to make mistakes so again be careful. Most of the steps are not all that bad but as mentioned in the process there are a couple of steps that we really need to be careful with since it is easy to make mistakes in those steps.
For all the functions that we are going to be looking at in this course if one is true then the other will also be true.
However, there are functions they are beyond the scope of this course however for which it is possible for only one of these to be true.
This is brought up because in all the problems here we will be just checking one of them. We just need to always remember that technically we should check both. However, it would be nice to actually start with this since we know what we should get.
- Inverse Functions
- Intro to inverse functions
This will work as a nice verification of the process. Here are the first few steps.
The next example can be a little messy so be careful with the work here. With this kind of problem it is very easy to make a mistake here.
Intro to inverse functions (video) | Khan Academy
That was a lot of work, but it all worked out in the end. We did all of our work correctly and we do in fact have the inverse. There is one final topic that we need to address quickly before we leave this section. There is an interesting relationship between the graph of a function and the graph of its inverse. Here is the graph of the function and inverse from the first two examples. 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 4 divided by 2 is is equal to x.
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 so now the range is now the domain for f inverse.
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 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. 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 so f of 0 is equal to 4.
Our function is mapping 0 to 4.
Properties of Inverse Functions
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 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.