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.