relationship between a function and its inverse - Mathematics Stack Exchange
Personal Finance · Home & Garden · Pets · Relationships · Sports · Religion When you're asked to draw a function and its inverse, you may choose to draw this see the points of the function reflecting over the line to become the inverse the function and then switch all x and y values in each point to graph the inverse. We will also discuss the process for finding an inverse function. and as noted in that section this means that there is a nice relationship between these two functions. . we plug different values into the function we get different function values. .. Most of the steps are not all that bad but as mentioned in the. Recall that to determine by the graph if an equation is a function, we have the vertical line test. If a vertical If the inverse is a function, each y value could only be paired with one x. Let's look Relation – a mapping of input values (x-values) onto output values (y-values). Here are 3 the inverse of a table: switch the x & y .
This function f is invertable, because it's a one-to-one mapping between the xs and the f of xs. No two xs map to the same f of x, so this is an invertable function. With that in mind, let's see if we can evaluate something like f inverse of 8. What is that going to be? I encourage you to pause the video and try to think about it.
Finding inverse functions: linear
So f of x, just as a reminder of what functions do, f of x is going to map from this domain, from a value in its domain to a corresponding value in the range. So this is what f does, this is domain Now f inverse, if you pass it, the value and the range, it'll map it back to the corresponding value in the domain.
But how do we think about it like this? Well, f inverse of 8, this is whatever maps to 8, so if this was 8, we'd have to say, well, what mapped to 8?
We see here f of 9 is 8, so f inverse of 8 is going to be equal to 9. If it makes it easier, we could construct a table, where I could say x and f inverse of x, and what I'd do is swap these two columns.
Inputs & outputs of inverse functions
All I did was swap these two. Now we're mapping from this to that. So f inverse of x is going to map from 7 to Notice, instead of mapping from this thing to that thing, we're now going to map from that thing to this thing. So f inverse is going to map from 13 to 5.
Calculus I - Inverse Functions
It's going to map from -7 to 6. It's going to map from 8 to 9, and it's going to map from 12 to Looks like I got all of them, yep. So all I did was swap these columns.
The f inverse maps from this column to that column. So I just swapped them out.Finding the inverse of a function
Now it becomes a little clearer. You see it right here, f inverse of 8, if you input 8 into f inverse, you get 9. Now we can use that to start doing fancier things. For the two functions that we started off this section with we could write either of the following two sets of notation. The process for finding the inverse of a function is a fairly simple one although there are a couple of steps that can on occasion be somewhat messy.
This is done to make the rest of the process easier.
This is the step where mistakes are most often made so be careful with this step. 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.
Finding inverse functions: linear (video) | Khan Academy
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.
That was a lot of work, but it all worked out in the end.