why do inverse functions only exist for one to one functions

Watch
rxrx2004
Badges: 15
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#1
Report Thread starter 3 months ago
#1
like y=x^2 is a many to one function, but it has an inverse as it can be reflected in the line y=x?
0
reply
mqb2766
Badges: 19
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#2
Report 3 months ago
#2
(Original post by rxrx2004)
like y=x^2 is a many to one function, but it has an inverse as it can be reflected in the line y=x?
At a very simple level you want the property


f^{-1}(f(x)) = x
Last edited by mqb2766; 3 months ago
0
reply
Pangol
Badges: 15
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#3
Report 3 months ago
#3
(Original post by rxrx2004)
like y=x^2 is a many to one function, but it has an inverse as it can be reflected in the line y=x?
Part of the definition of a function is that it has to be well-defined, in the sense that if you are given a value to plug in, you get one value out. In your example (but using functional notation), if f(x) = x2, then, for example, f(1) = 1 and f(-1) = 1. But what would be f-1(1)?
Last edited by Pangol; 3 months ago
0
reply
rxrx2004
Badges: 15
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#4
Report Thread starter 3 months ago
#4
(Original post by mqb2766)
At a very simple level you want the property


f^{-1}(f(x)) = x
but why

(Original post by Pangol)
Part of the definition of a function is that it has to be well-defined, in the sense that if you are given a value to plug in, you get one value out. In your example (but using functional notation), if f(x) = x2, then, for example, f(1) = 1 and f(-1) = 1. But what would be f-1(1)?
1 or -1
0
reply
mqb2766
Badges: 19
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#5
Report 3 months ago
#5
(Original post by rxrx2004)
but why
https://learn.lboro.ac.uk/archive/ol..._functions.pdf
Is a pretty good overview. A function (forwards or inverse, they're both functions) should evaluate to a single value. The inverse function is defined as per the previous post.

This is why
sqrt(2^2) = 2
i.e. its defined to be the positive root (single valued).
1
reply
RDKGames
Badges: 20
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#6
Report 3 months ago
#6
(Original post by rxrx2004)
1 or -1
Functions only have a single output.

You cannot say that f^{-1}(1) = \pm 1 because that’s two outputs … so f^{-1} is not an inverse function …
1
reply
Pangol
Badges: 15
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#7
Report 3 months ago
#7
(Original post by rxrx2004)
1 or -1
Well that's the problem. If f-1 were a function, you would be able to tell me the one value of f-1(1). But there isn't a single value, so it is not a function.
Last edited by Pangol; 3 months ago
0
reply
rxrx2004
Badges: 15
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#8
Report Thread starter 3 months ago
#8
(Original post by mqb2766)
https://learn.lboro.ac.uk/archive/ol..._functions.pdf
Is a pretty good overview. A function (forwards or inverse, they're both functions) should evaluate to a single value. The inverse function is defined as per the previous post.

This is why
sqrt(2^2) = 2
i.e. its defined to be the positive root (single valued).
ok thanks

(Original post by RDKGames)
Functions only have a single output.

You cannot say that f^{-1}(1) = \pm 1 because that’s two outputs … so f^{-1} is not an inverse function …
thanks

(Original post by Pangol)
Well that's the problem. If f-1 were a function, you would be able to tell me the one value of f-1(1). But there isn't a single value, so it is not a function.
thanks
0
reply
rxrx2004
Badges: 15
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#9
Report Thread starter 3 months ago
#9
(Original post by rxrx2004)
like y=x^2 is a many to one function, but it has an inverse as it can be reflected in the line y=x?
wait so y=x^2 isnt a function then?
Last edited by rxrx2004; 3 months ago
0
reply
mqb2766
Badges: 19
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#10
Report 3 months ago
#10
(Original post by rxrx2004)
wait so y=x^2 isnt a function then?
Yes it is, it evaluates to a single value.
its inverse however is only defined on the functions domain x>=0
Last edited by mqb2766; 3 months ago
0
reply
Pangol
Badges: 15
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#11
Report 3 months ago
#11
(Original post by rxrx2004)
wait so y=x^2 isnt a function then?
You don't run into the problem with f(x) = x2. For any value of x, there is only one value of x2.
0
reply
rxrx2004
Badges: 15
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#12
Report Thread starter 3 months ago
#12
(Original post by Pangol)
You don't run into the problem with f(x) = x2. For any value of x, there is only one value of x2.
(Original post by mqb2766)
Yes it is, it evaluates to a single value.
its inverse however is only defined on the functions domain x>=0
oh ok.
so y=root x isnt a function..i think i get it now
0
reply
mqb2766
Badges: 19
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#13
Report 3 months ago
#13
(Original post by rxrx2004)
oh ok.
so y=root x isnt a function..i think i get it now
It is a function, as its single valued. It returns positive values (and 0).
0
reply
Pangol
Badges: 15
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#14
Report 3 months ago
#14
(Original post by rxrx2004)
oh ok.
so y=root x isnt a function..i think i get it now
Well it is, because the definition of square root is to take the positive root. For example, the square root of 9 is 3, it's just the case that the square of negative 3 is also 9.
0
reply
rxrx2004
Badges: 15
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#15
Report Thread starter 3 months ago
#15
(Original post by Pangol)
Well it is, because the definition of square root is to take the positive root. For example, the square root of 9 is 3, it's just the case that the square of negative 3 is also 9.
but-but i always thought that you add a plus-minus sign at the front of a square root

(Original post by mqb2766)
It is a function, as its single valued. It returns positive values (and 0).
but if you reflect the graph of y=x^2 in the line y=x you'd get a graph for which it returns two values?
0
reply
mqb2766
Badges: 19
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#16
Report 3 months ago
#16
(Original post by rxrx2004)
but-but i always thought that you add a plus-minus sign at the front of a square root


but if you reflect the graph of y=x^2 in the line y=x you'd get a graph for which it returns two values?
The +/- reflects the fact that sqrt() returns a positive value.
You have to multiply it by -1 to get the corresponding negative root as it is defined to return the (single valued) positive value.

If it returned both values (it doesn't) there would be no need to write the +/-.
Last edited by mqb2766; 3 months ago
0
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Back
to top
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

Have you ever considered or are you currently considering an apprenticeship?

Yes, I am actively considering an apprenticeship (79)
13.57%
I am actively considering an alternative to uni that isn't an apprenticeship (11)
1.89%
I have considered an apprenticeship but it's not for me (147)
25.26%
I am considering a degree apprenticeship (47)
8.08%
I haven't considered an apprenticeship (280)
48.11%
Something else (let us know in the thread!) (18)
3.09%

Watched Threads

View All