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## Algebra 1

### Unit 8: Lesson 13

Intro to inverse functions# Inputs & outputs of inverse functions

CCSS.Math: ,

Sal explains that if f(a)=b, then f ⁻¹(b)=a, or in other words, the inverse function of f outputs a when its input is b.

## Video transcript

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.