Definition 1 (Inverse of a Function): This is the relation formed when the independent variable is exchanged with the dependent variable in a given relation.
This definition will be shown more clearly in the first example problem.
Definition 2 (Inverse Function): This is what the Inverse of a Function is verified in itself to be a function.
This will also be explained a bit more in the following example.
Let's start off with a fairly basic problem to cover this.
Example 1:
Consider the following function. Find it's inverse and verify.
a) Find the inverse
The first step to solve this is to make this function something a bit easier to understand. Something like this:
This simply replaces the function symbol with a y variable. Now it looks like an equation we know how to solve. The next thing to do is to make this equation an inverse of itself. This is where the Inverse of a Function definition comes into play.
This was done by exchanging the x and y variables. Now we just need to get y by itself.
That was easy enough right? Now that the equation is solved, just replace the y variable with our function symbol, now changed to show that it represents our function's inverse.
b) Verify
Simply finding the inverse of the function won't be enough though in this situation. We also need to verify that the inverse is a function too. We do this by using Composition of Functions. If you are unsure what that is, there is an earlier post to explain it more in-depth. The normal equation for composition of functions is this:
Since this is far easier for most people to visualize, we will use g to represent our inverse function. Visualize it like this:
Okay the first step of this part will be to set up our composition.
Now replace the g variable with our inverse equation's answer.
Now the next part is a little tricky to understand for some people. Consider that the inside of the parenthesis is the x variable in an equation. Now we need to take what our outside variable is (the F) and replace it with what it equals. In this case, all but the x value in the original function equation. Like this:
The parenthesis stays where it is because there is an x value in the function equation. We add the -6 because the function equation was x-6. Make sense? Now just simplify the rest.
Voila! Getting the result of =x is exactly what we are looking for! Though it seems like we have done a lot of work already, we aren't quite done. The good news is that the next part is almost exactly like what we did here. Now we have to reverse our starting point. Now our composition equation is like this:
All we have done is reverse our g and our F. We can't verify that the inverse itself is a function unless we try both options. Now let's do the same things we did with the previous group. I'm not going to walk though this time so if you get confused, refer back to the previous set.
And look at that! We get another =x. This shows that this is an Inverse Function. Refer to the definition above for the connection made here.
Since you will most definitely encounter more complex function problems than the one that we just completed, the following is a completed problem that uses a few more numbers. It isn't too much harder but you can use it to make sure you know how to solve these. There are small notes but if something really doesn't make sense, look back at the step-by-step with Example 1 to see what you are supposed to be doing.
Example 2:
Consider the following function. Find it's inverse and verify.
a) Find the inverse
Transfer to more understood variables
Flip the variables
Solve for y
Replace y
b) Verify
General term:
First section:
Second section:
If you were able to find the inverse of that function and verify it, great job! If not, maybe this video can help show things with a little different perspective. There are plenty of other great tools out there to help with learning this concept, but this is one that I found to be useful. Hopefully this post helped you learn how to work with Function Inverses!
