A very useful cousin of a function is called the function's inverse.
Do you remember a special type of function called a one-to-one function?
Well, what made it special? First of all, every element in the domain was paired
with a unique element in the range.
That's what made it a function, period. What makes it a one to one function is that
it works the other way as well.
It's a special type of function that every element in the range is paired
with a unique element in the domain.
In other words it works backwards. Every X is paired with only one y and
every y is paired with only one x. Now while a function is basically a path, if
you would, to get from x
to y, the inverse of a function is the way to get back.
I say, to get back from y, back to X. Now consider the reason we're talking about
one-to-one functions is unless you're working with a one-to-one function that
works both ways, if you would,
there isn't a way back. So you only have an inverse of a function or a function
only has an inverse if it is in fact originally a one-to-one function.
Otherwise, there's no way back. Anyway,
now this relation,
is a function; it gets the function seal of approval.
because no x is paired with more than one y.
The inverse of it would be just flipping the domain and the range.
It's very simple but that's the idea of the inverse. The inverse is going to map
the range back to the domain or the y's back to the x's.
Now this isn't always that simple.
Let's look at this relation. Now every X is paired with only one
y. So it is a function;
it does get the function seal of approval. All the x's are in fact
different.
If we tried to find the inverse of it, basically by flipping the X's and the
Y's,
Well, in this case, all the x's aren't different.
So, while it would have been an inverse, it's not an inverse function and the
reason for that, the inverse isn't a function is because the original
function wasn't a one-to-one function, was it? It was a function but it wasn't a
one-to-one function. So we'll have to start with a one-to-one function in the
first place to even have an inverse ever. Here is
how we find the inverse formula of a function.
There's four steps. A lot of this is just notation changes. For instance, the first
step is to replace the f of X with what it really is the Y. Then we're actually
going to interchange the x and the y in the equation. Flip them and then here's
something you should be good at... We're going to solve for y.
Once we've solved for y and we've gotten y by itself,
we're going to replace that y with the notation for F inverse of X and that's
written f minus 1 of X. It doesn't mean to the negative 1 power.
It means the inverse function of X.
[I don't understand. ] Ok, well, let's try one. We're going to try and find the inverse
of this function f of x. The first thing we'll do is replace f of x with y.
Okay, there you go. Second step, interchange the X and the y so I'll
write the function normally except I'll interchange the x and y.
Ok, that's the second step. Now the third step is to do whatever it takes to solve
for y; in other words get y by itself.
Now, who's keeping y from being by itself? If you look at him, the four is. So I'll
subtract 4 from both sides
and y will equal x minus 4.
That's basically the inverse function. But for notation sake
the fourth step is to replace the y part with the notation for that F inverse of
x and that's the inverse. That's the inverse function of X in this case.
Recall that the original function will take any X that you put ,in any value for
x and pair it with a new y. I would like to say map
every X to its own unique y. The inverse function is going to get you
back from that y, back to x.
Interestingly enough you can test whether you're correct, whether you've
found the correct inverse function by doing something that we just covered
called compositions.
Ok, if we compose the inverse function with the original function or f inverse
of f of x it should bring us back to x. and let's try that.
[Let's play here we go.]
We're going to find the composition of the inverse of f of X. Ok so we're going
to take the inverse of the original function. Now the original, I'm sorry, the
inverse function is X minus 4 so we're going to replace X minus four, the X
portion of that, with the original function.
I'd like to use parentheses here and now I'll simplify and look what I end up
getting back to the mother lode
x that's that's how I'll know. I check if I can compose f inverse with f of X and
I get back to X. Well, then I know I have the right inverse and cool as a mule.
You know what, if we actually graph any function and on the same picture, graph
its inverse. Iwant to show you what happens when we graph the original
function.
I'll get three easy points. We plug in 0, 1 and negative 1 because they'll fit on
the graph. I put in a 0, I get four. I put in a 1, I get five and if I put a
negative 1, I get three.
I graph these three points. They line up and there's the original function,
y or f of x equals x plus 4. Now if I do the inverse function, try the same
three points, 0 negative 0, 1 and negative 1, I get (0,negative 4), (1, negative 3) and
(negative 1, negative 5) and if I graph those, they also line up and the inverse
function in blue their graphs right there
Now, what will always happen when you graph a function and it's inverse
function since one is the, if you would, interchanging of x and y is that they
will be a mirror of each other about the line y equals x.
I could actually fold this graph on the dotted line which is that the graph of
the line y equals x and the red would end up on top of the blue. That will
always happen.
Ok so the graphs of inverse functions always mirror each other about the line
y equals x.
That's another test that you could use. On your calculator, even, let's try
another.
We're going to find the inverse of this function, the function f of x equals 3x
plus 2.
We need to find out what function would get us back from y back to x.
I, well, remember the steps! Replace f of X with Y because that's what it really is
and then rewrite the whole equation interchanging x and y.
Ok, now solve the equation for y; get y by itself.
Now I have two things keeping y from being by itself,
the two and the three. Let me subtract two from both sides
and I'll get x minus 2 there. Can't, can't do anything there. Now divide both sides
by 3 to undo the 3 and y will equal x minus 2 over 3.
Now the last step is to replace the Y with the notation F inverse of X. This
doesn't surprise me much that the inverse of multiplying by 3 and adding
to the inverse of that is subtracting 2 and dividing by 3.
It's just the opposite operations, isn't it?
So there's your inverse function. Now let's
test it out
not because I said so but just to practice with compositions. Remember that
if we compose each we should end up back at x.
The composition of f , uh, inverse of x and f of x should get me back to x.
Let's try it. There's the inverse function and I'm going to replace X with
the original function 3x plus 2.
Now let me simplify. The twos cancel and the threes cancel and what's left?
There he is, x. We got back to X. I must have found the correct inverse function
Pretty cool.
Let's do the other test.
Let's graph both the original function and it's inverse and see if it mirrors
about the line y equals x. The original function put in these three values and
I'll get these three points. Now it's a little bit different than our picture
before but they do line up. Now the inverse function and we get three points
for him and let's see me graph those points, hmm... looks different.
They do line up and, son of a gun, if you draw the line y equals x and if you fold it
on that dotted line that red would end up on top of that blue.
So the graphs of inverse functions always mirror each other about the line
y equals x meaning that you could fold your graph over the line y equals x on
the dotted line and they fall on top of each other.
Ok, let's go practice it. Go do that homework.