You may by now be familiar with the notion
of evaluating a function with a particular value,
so for example, if this table is our function definition,
if someone were to say, "Well, what is f of -9?"
you could say, okay, if we input -9 into our function,
if x is -9, this table tell us that f of x is going to be equal to 5.
You might already have experience with doing composite functions,
where you say, f of f of -9 plus 1.
So this is interesting, it seems very daunting,
but you say, well we know what f of -9 is, this is going to be 5,
so it's going to be f of 5 plus 1.
So this is going to be equal to f of 6,
and if we look at our table, f of 6 is equal to -7.
So all of that is review so far,
but what I want to now do is start evaluating the inverse of functions.
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.
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...
and this right over here is the range.
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.
f of x goes from -9 to 5, f inverse of x goes from 5 to -9.
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 -7.
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.
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 11.
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. 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.
We can evaluate something like f of f inverse of 7.
f of f inverse of 7.
What is this going to be?
Let's first evaluate f inverse of 7.
f inverse of 7 maps from 7 to -7.
So this is going to be f of this stuff in here,
f inverse of 7, you see, is -7.
And then to evaluate the function, f of -7 is going to be 7.
And that makes complete sense.
We mapped from f inverse of 7 to -7
and evaluating the function of that, went back to 7.
So let's do one more of these just to really feel comfortable
with mapping back-and-forth between these two sets,
between applying the function and the inverse of the function.
Let's evaluate f inverse of f inverse of 13.
f inverse of 13.
What is that going to be?
I encourage you to pause the video and try to figure it out.
What's f inverse of 13?
That's, looking at this table right here, f inverse goes from 13 to 5.
You see it over here, f went from 5 to 13, so f inverse is going to go from 13 to 5.
So, f inverse of 13 is going to be 5,
so this is the same thing as f inverse of 5.
And f inverse of 5? -9. So this is going to be equal to -9.
Once again, f inverse goes from 5 to -9.
So at first when you start doing these functions and inverse of functions
it looks a little confusing, hey, I'm going back and forth,
but you just have to remember
a function maps from one set of numbers to another set of numbers.
The inverse of that function goes the other way.
If the function goes from 9 to 8, the inverse is going to go from 8 to 9.
So one way to think about it is, you just switch these columns.
Hopefully, that clarifies more things than it confuses.