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.

Should f(f-1(7)) = 6 ?

What program does he use?

Hey sal were you forced to do tons of work when you were younger like I am since I have Asian parents

Can't it go back to 13

how do i do this problem

g(x)=2(x+6)-13?

I love you

Lolol!

THANK YOU SOOO MUCH! I'm an autodidact using Sheldon Axler's Precalculus book and I must say, I don't think it's as clear as it could be, so thanks again for clarifying something pretty basic!

I think someone tried to italicize parts of the Closed Captions without realizing they don't execute HTML because variables are surrounded by <i> </i> even though Sal is not saying that.

lololololololollolololololololollolol

When am I going to use this in my life seriously someone tell me

Yep this Is pointless. Math is useless whoever invented math or how this came up should've been hit on the head with a rock.

stop reading comments and go back to studying!!!

OH MY GOSH THANK YOU SOOOOOOOOOOO MUCH!!!!!!!!!

This is amazing

Thanks