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### Course: Algebra (all content)>Unit 7

Lesson 19: Introduction to inverses of functions (Algebra 2 level)

# 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!
• 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
• @ 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.
• 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.
• @ I don't understand how you got the inverse of 7.
• 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.
• What is many-to-one and one-to-one?
• 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.
• 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?
• 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.
• 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.
• 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 ?
• 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.
• Would f(f^1(x)) always equal x because both functions cancel each other out
• 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.
• 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?