Edexcel C3 Functions question, many-one and one-one

Hello, I would really appreciate if one of you could explain many-one or one-one when answering questions like why does this function not have an inverse, or something similar, I don't understand it. I tried looking this up online but just couldn't find something good that explains it well. I think it is always best to have something explained by another student.

Thank you
Reda

Reply 1

Zacken

22

Original post by Reda2

Hello, I would really appreciate if one of you could explain many-one or one-one when answering questions like why does this function not have an inverse, or something similar, I don't understand it. I tried looking this up online but just couldn't find something good that explains it well. I think it is always best to have something explained by another student.

Thank you
Reda

So functions take in an input and give out an output, yeah?

The inverse functions allows me to give an output and the inverse function will tell me precisely which input gives me that output.

You have two kinds of function, one is one-to-one, i.e: for every input, there is exactly one output.

For example, the cube function (

f(x) = x^3

) is such an example:

If I give it a number, let's say

2

- it gives out an output

2^3 = 8

. No other number I put in will give me 8. This is a one to one function. Every input gives one output.

Hence, since it is one to one, an inverse function will exist. Namely the cube root function.

f^{-1}(x) = \sqrt[3]{x}

. If I give it an output tex]x = 8, it tells me that the input I put into my cube function was

\sqrt[3]{8} = 2

.

However, some other functions are many-to-one, that is, if I give it an input, it will only give out one output, yes - this is true. But one output can come from two differnent inputs. So for example, the cube function was not an many-to-one because 8 corresponds to only only 2, there is no other number I could put in that gives me 8 as well.

However, some other functions do have this property that the outputs could come from different numbers. Such an example is the squaring function (

f(x) = x^2

):

This function has the property that any output comes from two different inputs. For example, the output 4 comes from

2^2 = 4

but it also comes from

(-2)^2 = 4

. So, with that in mind - how on earth are we going to define an inverse function?! What possible function could I say "where does 4 come from?" and then that function then tells me that the 4 comes from 2 or -2? How on earth is the function meant to know whether the input that gives 4 is -2 or 2? It won't, hence the inverse just doesn't exist.

So, to recap: a one-to-one function means that the inverse exists because for any output, there's only one possible number that gives leads to that output. Hence I can give you an inverse function.

However, a one-to-many function can't have an inverse because if I give it a output, it won't know which input that number comes from. That is, for any output, there is more than 1 number/input that gives that output and hence I can't give you an inverse function to find that input.

Reply 2

KINGYusuf

12

Original post by Zacken

So functions take in an input and give out an output, yeah?

The inverse functions allows me to give an output and the inverse function will tell me precisely which input gives me that output.

You have two kinds of function, one is one-to-one, i.e: for every input, there is exactly one output.

For example, the cube function (

f(x) = x^3

) is such an example:

If I give it a number, let's say

2

- it gives out an output

2^3 = 8

. No other number I put in will give me 8. This is a one to one function. Every input gives one output.

Hence, since it is one to one, an inverse function will exist. Namely the cube root function.

f^{-1}(x) = \sqrt[3]{x}

. If I give it an output tex]x = 8

, it tells me that the input I put into my cube function was

\sqrt[3]{8} = 2

.

However, some other functions are many-to-one, that is, if I give it an input, it will only give out one output, yes - this is true. But one output can come from two differnent inputs. So for example, the cube function was not an many-to-one because 8 corresponds to only only 2, there is no other number I could put in that gives me 8 as well.

However, some other functions do have this property that the outputs could come from different numbers. Such an example is the squaring function (

f(x) = x^2

):

This function has the property that any output comes from two different inputs. For example, the output 4 comes from

2^2 = 4

but it also comes from

(-2)^2 = 4

. So, with that in mind - how on earth are we going to define an inverse function?! What possible function could I say "where does 4 come from?" and then that function then tells me that the 4 comes from 2 or -2? How on earth is the function meant to know whether the input that gives 4 is -2 or 2? It won't, hence the inverse just doesn't exist.

So, to recap: a one-to-one function means that the inverse exists because for any output, there's only one possible number that gives leads to that output. Hence I can give you an inverse function.

However, a one-to-many function can't have an inverse because if I give it a output, it won't know which input that number comes from. That is, for any output, there is more than 1 number/input that gives that output and hence I can't give you an inverse function to find that input.

Oh Zacken you beautiful human

Reply 3

Zacken

22

Original post by KINGYusuf

Oh Zacken you beautiful human

I take it that you've found it helpful? :smile:

Reply 4

Notnek

21

Original post by Zacken

...

Perfect explanation.

Reply 5

Zacken

22

Original post by notnek

Perfect explanation.

Cheers.

Reply 6

Reda2

OP

14

Original post by Zacken

So functions take in an input and give out an output, yeah?

The inverse functions allows me to give an output and the inverse function will tell me precisely which input gives me that output.

You have two kinds of function, one is one-to-one, i.e: for every input, there is exactly one output.

For example, the cube function (

f(x) = x^3

) is such an example:

If I give it a number, let's say

2

- it gives out an output

2^3 = 8

. No other number I put in will give me 8. This is a one to one function. Every input gives one output.

Hence, since it is one to one, an inverse function will exist. Namely the cube root function.

f^{-1}(x) = \sqrt[3]{x}

. If I give it an output tex]x = 8

, it tells me that the input I put into my cube function was

\sqrt[3]{8} = 2

.

However, some other functions are many-to-one, that is, if I give it an input, it will only give out one output, yes - this is true. But one output can come from two differnent inputs. So for example, the cube function was not an many-to-one because 8 corresponds to only only 2, there is no other number I could put in that gives me 8 as well.

However, some other functions do have this property that the outputs could come from different numbers. Such an example is the squaring function (

f(x) = x^2

):

This function has the property that any output comes from two different inputs. For example, the output 4 comes from

2^2 = 4

but it also comes from

(-2)^2 = 4

. So, with that in mind - how on earth are we going to define an inverse function?! What possible function could I say "where does 4 come from?" and then that function then tells me that the 4 comes from 2 or -2? How on earth is the function meant to know whether the input that gives 4 is -2 or 2? It won't, hence the inverse just doesn't exist.

So, to recap: a one-to-one function means that the inverse exists because for any output, there's only one possible number that gives leads to that output. Hence I can give you an inverse function.

However, a one-to-many function can't have an inverse because if I give it a output, it won't know which input that number comes from. That is, for any output, there is more than 1 number/input that gives that output and hence I can't give you an inverse function to find that input.

What king said... Thank you so much mate, you explained it well :smile:

Reply 7

Zacken

22