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functions and inverses

My text book says that one-one functions and many-one functions have inverses, however the exam answers say that a function does not have a inverse because it is not one-one. Both of these cant be right?

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Reply 1
I know if you restrict domain of many-1 function is becomes 1-1 and has a inverse but does a unrestricted many-1 have a inverse? Or would the inverse make it 1-many??
(edited 11 years ago)
Reply 2
Original post by Lay-Z
My text book says that one-one functions and many-one functions have inverses, however the exam answers say that a function does not have a inverse because it is not one-one. Both of these cant be right?


It's likely you've missed a key point in your textbook. A many to one function can have an inverse PROVIDED that the domain is restricted to some values of x.
Reply 3
Original post by Madalaine M
It's likely you've missed a key point in your textbook. A many to one function can have an inverse PROVIDED that the domain is restricted to some values of x.

so a unrestricted many-1 function has no inverse?
Original post by Lay-Z
so a unrestricted many-1 function has no inverse?


Correct.
Reply 5
Only a one to one function which is surjective (aka onto i.e. hits everything in the codomain/range) can have an inverse. Think about it this way the function is a machine that for every input in the the domain gives an output in the codomain i.e. the function f(x) = 2 x from real numbers to real numbers gives you a number after you give it a number, i.e. after you give it 4 it gives you 8.

An inverse is a machine like this working in reverse i.e you give it something (y say) in the codomain of your first function and it gives you what you would need too put in the original machine to get y. If f(x) = 2x you can do this because it is 1-1 and onto so for each real number y if you fed y/2 into the machine you would just get y back.

But if we take g(x) = x^2 and try to invert it, we ask what did we started with if we get given a 4 by the machine... the problem is you cant actually tell because it could be +2 or -2 and the rules a a function is it can only give you one value so you cand make a machine to invert g(x)=x^2 properly.

Whats worse is if we ask what did we start with if we get -1 from the machine the answer is nothing but a function has to give an output for every input so the fact that g(x)=x^2 doesn't hit every real number is a very important point as to why it does not have an inverse.
(edited 11 years ago)
Reply 6
Original post by Dadeyemi
Only a one to one function which is surjective (aka onto i.e. hits everything in the codomain/range) can have an inverse. Think about it this way the function is a machine that for every input in the the domain gives an output in the codomain i.e. the function f(x) = 2 x from real numbers to real numbers gives you a number after you give it a number, i.e. after you give it 4 it gives you 8.

An inverse is a machine like this working in reverse i.e you give it something (y say) in the codomain of your first function and it gives you what you would need too put in the original machine to get y. If f(x) = 2x you can do this because it is 1-1 and onto so for each real number y if you fed y/2 into the machine you would just get y back.

But if we take g(x) = x^2 and try to invert it, we ask what did we started with if we get given a 4 by the machine... the problem is you cant actually tell because it could be +2 or -2 and the rules a a function is it can only give you one value so you cand make a machine to invert g(x)=x^2 properly.

Whats worse is if we ask what did we start with if we get -1 from the machine the answer is nothing but a function has to give an output for every input so the fact that g(x)=x^2 doesn't hit every real number is a very important point as to why it does not have an inverse.

How is this a function then, if a function must have a inverse to be classed as a function? This is a unrestricted domain and many-to-one yet it is saying it is a functionUntitled.png? Unless a function is not defined as having a inverse, which is what we are taught.
Reply 7
Original post by Lay-Z
How is this a function then, if a function must have a inverse to be classed as a function? This is a unrestricted domain and many-to-one yet it is saying it is a functionUntitled.png? Unless a function is not defined as having a inverse, which is what we are taught.


What makes you think a function "must have an inverse"?
Reply 8
Original post by davros
What makes you think a function "must have an inverse"?

How would you define a function then?
Reply 9
Original post by Lay-Z
How would you define a function then?


A function is a mapping from set A to set B, say, which associates a unique element of set B which any element of set A.

If you picture this as a set of lines drawn from the elements of A to those of B, then for a function, you're not allowed to have more than one line emerging from any given element of A.

You are allowed, however, to have more than one line going to any given element of B.

For example, the mapping given by the rule f(x)=x2f(x) = x^2 where xRx \in \mathbb{R} is a function, since no real number has more than one square (e.g. the square of 3 is only 9, and the square of -3 is only 9). Thus, if we drew this mapping using lines between sets, we'd never see two lines emerging from one element of set A (which in this case is R\mathbb{R}) and going to different elements in set B (which in this case is also R\mathbb{R}).

However, f(x)=x2f(x) = x^2 doesn't have an inverse function i.e. a mapping that takes us back from set B to set A, with the necessary property. That's because 9 , for example, as we have seen, is mapped to by both 3 and -3, so it has two lines coming to it. If we try to create a mapping that takes us back from B to A, we can't make it a function (i.e. a mapping with only one line emerging from any element) unless we discard some of the lines.

For example, we could decide to discard all of the lines that take us back to negative elements of set A, so that we'd only have a line from 9 to 3, say. In this case, we would do this by restricting the domain of f(x)f(x) to be the positive real numbers; then the inverse function f1(x)f^{-1}(x) would be defined.
Reply 10
Original post by Lay-Z
How would you define a function then?



Original post by atsruser
A function is a mapping from set A to set B, say, which associates a unique element of set B which any element of set A.

If you picture this as a set of lines drawn from the elements of A to those of B, then for a function, you're not allowed to have more than one line emerging from any given element of A.

You are allowed, however, to have more than one line going to any given element of B.

For example, the mapping given by the rule f(x)=x2f(x) = x^2 where xRx \in \mathbb{R} is a function, since no real number has more than one square (e.g. the square of 3 is only 9, and the square of -3 is only 9). Thus, if we drew this mapping using lines between sets, we'd never see two lines emerging from one element of set A (which in this case is R\mathbb{R}) and going to different elements in set B (which in this case is also R\mathbb{R}).

However, f(x)=x2f(x) = x^2 doesn't have an inverse function i.e. a mapping that takes us back from set B to set A, with the necessary property. That's because 9 , for example, as we have seen, is mapped to by both 3 and -3, so it has two lines coming to it. If we try to create a mapping that takes us back from B to A, we can't make it a function (i.e. a mapping with only one line emerging from any element) unless we discard some of the lines.

For example, we could decide to discard all of the lines that take us back to negative elements of set A, so that we'd only have a line from 9 to 3, say. In this case, we would do this by restricting the domain of f(x)f(x) to be the positive real numbers; then the inverse function f1(x)f^{-1}(x) would be defined.


Basically this :smile:

A function is just a mapping that takes one value to a another unique value, and need not have an inverse. If it were a requirement for a function to have an inverse, then there would be no point having questions that say "does this function have an inverse"?
Reply 11
Original post by Lay-Z
How is this a function then, if a function must have a inverse to be classed as a function? This is a unrestricted domain and many-to-one yet it is saying it is a functionUntitled.png? Unless a function is not defined as having a inverse, which is what we are taught.


You were taught wrong, thats actually a little worrying, what exam board is this?
(edited 11 years ago)
Reply 12
Original post by Dadeyemi
You were taught wrong, thats actually a little worrying, what exam board is this?

AQA
Reply 13
Original post by Lay-Z
AQA


Hmm I would advise you check that definition your were given with your teacher next time you have a chance to, I don't think that's the definition the exam board uses (and I'm quite positive its not the standard one).
(edited 11 years ago)
Reply 14
Original post by Dadeyemi
Hmm I would advise you check that definition your were given with your teacher next time you have a chance to, I don't think that's the definition the exam board uses (and I'm quite positive its not the standard one).


Alright

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Reply 15
The way they teach about functions in school is so half assed it is no wonder that it confuses more than enlightens...
Reply 16
Original post by Mark85
The way they teach about functions in school is so half assed it is no wonder that it confuses more than enlightens...


Replace the word "functions" with just about any aspect of Mathematics and you may be onto something. (Actually I hear they teach strange things about electron valency at GCSE, and French is now taught by selecting multiple choice answers and filling in missing words in passages of text.)
Original post by davros
Replace the word "functions" with just about any aspect of Mathematics and you may be onto something. (Actually I hear they teach strange things about electron valency at GCSE, and French is now taught by selecting multiple choice answers and filling in missing words in passages of text.)


How would you teach maths, or how do you teach maths even?
At every stage we address the limitations of the syllabus at the previous stage

Yes, there numbers less than 0
Yes, you can divide a small number by a big number
No, multiplying does not always give a bigger number
Yes, actually, you can square root a negative number
Sqrt{2} is a better answer than 1.414....


ETC ETC ETC

it i no just from Level 3 to level 4
Reply 19
Original post by TenOfThem
At every stage we address the limitations of the syllabus at the previous stage

Yes, there numbers less than 0
Yes, you can divide a small number by a big number
No, multiplying does not always give a bigger number
Yes, actually, you can square root a negative number
Sqrt{2} is a better answer than 1.414....


ETC ETC ETC

it i no just from Level 3 to level 4


In the case of functions - this is done too soon. There is too much emphasis on the definition of a function before the need for it.

Then, when it is taught - there is this whole business about still teaching that a function is defined by a 'rule' and that the domain of a real function is the largest for which the rule makes sense and the codomain is basically completely unemphasised. This serves to obfuscate more than clarify the essential points.

At the end of the day, if you want to introduce the notion of function to discuss the possibility of inverses, you need to state straight off the bat that the domain and codomain are part of the essential data of a function rather than the wishy-washy crap that you see in text books (or questions on here from people confused by said textbooks).

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