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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.

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  • aqualine ultimate style avatar for user Tullio
    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!
    (64 votes)
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  • blobby green style avatar for user Catherine Chalkley
    @ the 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.
    (19 votes)
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  • leafers tree style avatar for user jazzybenigno11
    @ I don't understand how you got the inverse of 7.
    (7 votes)
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  • duskpin sapling style avatar for user tushar31gupta
    What is many-to-one and one-to-one?
    (5 votes)
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    • aqualine tree style avatar for user Judith Gibson
      At about to , 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.
      (9 votes)
  • starky ultimate style avatar for user Rob611
    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?
    (6 votes)
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    • mr pink green style avatar for user David Severin
      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.
      (5 votes)
  • blobby green style avatar for user William Mitchell
    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)
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  • hopper cool style avatar for user Yash
    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)
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    • eggleston green style avatar for user Ants Work Diligently
      - 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)
  • aqualine ultimate style avatar for user Zen
    How would you find a function that was not on the table while keeping the input and output balanced?
    (3 votes)
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  • leafers ultimate style avatar for user lizsol35
    But what if many domains have the same range?
    (1 vote)
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    • starky tree style avatar for user na609347
      it is convention that many inputs have same output like many domains have same range.
      for example a b and c persons are buying same thing from a shop then the seller will give the same thing to person a, b, and c. If the seller will not confused then how the function can be..
      (3 votes)
  • blobby green style avatar for user vani.v.sethi
    How to find f-1(x). When f(x)= 3+2x ?
    (2 votes)
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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.