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### Course: Get ready for Precalculus > Unit 3

Lesson 6: Intro to inverse functions# Inputs & outputs of inverse functions

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.

## Want to join the conversation?

- I have a doubt. An x in the domain of a function is mapped to just one y in the range. But y could be mapped from more than one x. So, what is the result of the inverse of a function when you input a y that could be mapped to more x? Is it possible? Will the function return more results? Or do we just swap x and y and do we still have more x mapping to a single y? Thanks for your time!(74 votes)
- Hi, Tullio

If y could be mapped from more than one x, that's not a function anymore.

So we can not have an inverse function if we don't have an inverse function in the first place.

Hope it helps(7 votes)

- @6:05the result of f^-1(f^-1(13)) was found to be 9. From f^-1(13)=5 and then f^-1(5) = 9. I am confused because I thought that it would have been 13 again, just as the inverse of 7 example. I am having a hard time reconciling this issue.(21 votes)
- f^-1 maps the function from input to output. When you said that it would be 13 again, you were assuming that it would go output to input after the first functions. That is incorrect because the question is asking for input to output again after the first function.(10 votes)

- @4:30I don't understand how you got the inverse of 7.(6 votes)
- All Sal really did was look for an input (an "x value") that would give him a function value of 7. If you look at the orange/brown table you can see that a function value of 7 occurs when the input is -7.(8 votes)

- What is many-to-one and one-to-one?(4 votes)
- At about0:53to1:10, Sal mentions that the function is a one-to-one mapping, since no two x's map to the same f(x). This means the function is invertible (has an inverse).

So if we're given a value for f(x), we will know without doubt which value of x produced it.

A many-to-one mapping means that at least two values of x (and maybe more) map to a single value of f(x).

So if we were given a value of f(x) to start with, we wouldn't be able to say with certainty which value of x had produced it.(11 votes)

- Is it possible to find F^-1(f(58)) with a table of

x 5, 3, 1, 18, 0, 9

f(x) 9, -2, -5, -1, 1, 11?

I don't think it is but I was asked this in a problem and was wondering if this could a mistake.

When I looked at the answer it said it was 58. Why is this?(5 votes)- This is true by definition of inverse. f(58) would lend an answer of (58,y) depending on the function. It really does not matter what y is. The inverse of this function would have the x and y places change, so f-1(f(58)) would have this point at (y,58), so it would map right back to 58.

So try it with a simple equation and its inverse. If f(x)=2x + 3, inverse would be found by x=2y+3, subtract 3 to get x-3 = 2y, divide by 2 to get y = (x-3)/2. Lets find f-1(f(4)). f94) = 2(4)+3 = 11. So f-1(11) = (11-3)/2 = 8/2= 4.

or f-1(f(-5) f(-5) = 2(-5) + 3 = -7, f-1(-7) = (-7-3)/2 = -10/2 = -5. Try f-1(f(58)). f(58) = 2(58)+3=119. f-1(119) = (119-3)/2 = 116/2 = 58. So the table is irrelevant to the question, it would work for any function.(8 votes)

- This explanation is so much clearer than the intro to inverse functions. The key here is the tables, and that the roles of x and y are reversed. That explains why the inverse graph does not overlay the original function graph in the intro.(6 votes)
- what if we had the table below:

x |f(x)

1 | 3

3 | 5

7 | 9

8 | 5

as you see 5 is repeating for different x values {3,8}.

in this case is f^-1(5) undefined ?(3 votes)- Your function has no inverse function. Functions only have an inverse if there is a one-one relationship between X and Y. Since your functions has a two-one relationship, it has no inverse.

Sometimes to force an inverse, we restrict the domain (acceptable input values) for the original function. This is commonly done with quadratics.(3 votes)

- what if one of the values isn't on the table, for example, i have the same exact problem as Sal, but it is Inverse
*f ( f*(576)) ? so how would you do this?(3 votes)- - Inverse functions have an output and trace it to an input.

- The output that you are trying to find an input for is: "f(576)"

- This represents an output, but it is phrased like

"The output for input 576."

From here it is very easy to find what input you had in the first place (576), since the input is used as part of the output.(1 vote)

- How would you find a function that was not on the table while keeping the input and output balanced?(3 votes)
- Would f(f^1(x)) always equal x because both functions cancel each other out(2 votes)
- If f(x) and f^-1(x) are inverses, then...

f(f^-1(x)) = x AND f-1(f(x)) = x

Both must be true for the functions to be inverses. And, yes, they equal x because the original function and its inverse cancel out the operations performed by each individually.(3 votes)

## 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.