Professor Dave here, let’s talk about inverse functions.
We’ve learned that when you plug a number into a function, it spits out some other number.
Now we want to learn about inverse functions.
If a function gives you some number, if you then plug that number into the inverse function
of that original function, we should get the number we started with.
In this way, the word inverse is being used in a different context than when we say the
inverse of a fraction.
Inverse functions are not just reciprocals of the original function.
Instead, we must do some algebra to find the inverse of a particular function, but we will
always use the same approach.
We simply allow F of X, which we can also express as Y, to swap places with X, and then
we solve for Y.
For example, if we have Y equals X plus two, then these swap places, we subtract two from
both sides, and Y is equal to X minus two.
This is a totally new F of X, which we express as F inverse of X, with a negative one superscript
after the F.
This makes sense, because the inverse function must undo every operation that the original
function did.
So if F of X adds two to any input, F inverse of X must subtract two from any input.
We can prove this works by evaluating for some X, like three.
Plugging three into F, we get five.
Plugging five into F inverse, we get three, which is what we started with.
Before we practice more of these, let’s point out just a few things about inverse
functions.
We should note that the domain of F is equal to the range of F inverse, and the domain
of F inverse is equal to the range of F. In other words, if F is the set of ordered pairs
XY, then F inverse is the set of ordered pairs YX.
This is evidenced by the fact that we switch the positions of X and Y when finding the
inverse function.
If after we switch them, we find that we are unable to solve for Y, then the function does
not have an inverse.
If we are able to solve, it does have an inverse.
If an inverse function exists, we can demonstrate this for any X input, applying F and then
F inverse to get the original value, but it should also be the case that F of F inverse
of X equals X, and also that F inverse of F of X equals X, which is a good way to check
your math.
Let’s show this for the previous example.
F of X is X plus two.
If we plug F inverse in there, or X minus two, we get X minus two plus two.
This is equal to X.
The same goes for the reverse operation.
Now that we understand the relationship between a function and its inverse function, we can
try some trickier examples.
Let’s say we have F of X equals four X minus five.
Again, we change this to a Y, swap places with X, and solve.
We quickly see that we must add five, and then divide by four, leaving us with X plus
five over four as the inverse function.
We could pick a value and evaluate, or plug one function into the other to see that these
are indeed inverse functions, as each one systematically undoes each operation in the other.
What about X cubed plus one?
Again, swap the variables and solve.
Subtract one, take the cube root, and we get the cube root of X minus one.
So we can see that finding the inverse function will typically just involve simple algebra.
One other thing we want to point out about inverse functions, is that since every point
on one will correspond to a point on the other but with the coordinates reversed, meaning
that every XY on one becomes YX on the other, it is the case that to get the inverse of
a function, we can simply reflect the function across the line Y equals X.
This makes sense, because the inverse of Y equals X is Y equals X, so that won’t go
anywhere, but something like this function will have all of its X’s turn into Y’s,
and all of its Y’s turn into X’s, so we get this instead.
We can see that all the points are accounted for, just with their coordinates reversed.
So now that we know how to find inverse functions, let’s check comprehension.