Why this is not a function? x -> distance from origin to point (0,x)

[Masaru]

I have come across the following basic question about "Function":

Determine whether or not this is a function:

"x" is mapped to "the length of the line from the origin to (0, x) where x is Real.

And the answer provided is as follows:

Not a function, undefined when x < 0

I do not understand why it is not a function because, for example,

When x = 1, the length of the line from the origin to (0, 1) is 1.
When x = - 1,

the length of the line from the origin to (0, -1) is also 1

So, to generalise it, when x = k,

the length of the line from the origin to (0, k) is k.​

​

And w

hen x = - k, the length of the line from the origin to (0, -k) is also k.

Therefore, it is a rule that maps a single number to another single number for a defined set of input numbers, and this is an example of many-to-one functions like y = x^2.

Am I wrong?

Please help me to understand why this is not a function.
​

 

[tkhunny]

y = |x| is a Function. Your example is very similar.

Perhaps the author is thinking the distance from 0 to your -k is negative and that isn't a distance.

 

[stapel]

Determine whether or not this is a function:

"x" is mapped to "the length of the line from the origin to (0, x) where x is Real.

And the answer provided is as follows:

Not a function, undefined when x < 0

I do not understand why it is not a function....

I kinda agree with you. There is nothing in the definition that says the x-value can't be negative, because the point (0, x) for x < 0 is well-defined, as is the length of the line segment from the origin, down the y-axis, to (0, x), that length being |x| = -x. (Here, the "minus" means "change the sign", not "negative". Because x started out negative in this case, -x is then positive.)

How touchy is your instructor? Would s/he be open to questioning this answer...?

 

[Dr.Peterson]

I can't think of any good reason for saying the distance is ever undefined.

Can you show us the context of the question? If it is a book, you might post a picture of the page; if it is a web site, give us the URL. There may be something in the context that would change our answers, especially if you have omitted part of the problem.

 

[Masaru]

The Picture of the Page (Questions & Answers)

I can't think of any good reason for saying the distance is ever undefined.

Can you show us the context of the question? If it is a book, you might post a picture of the page; if it is a web site, give us the URL. There may be something in the context that would change our answers, especially if you have omitted part of the problem.

I have attached the pictures of the pages for the questions and answers from the book.
As you can see, the question and its answer that I am struggling with is a part of a set of questions.
Having said that, this question is not related to previous questions or anything as you can see.

For your convenience, I copied the questions and answers from the book below - the same as the content of the attached pictures:

Question: Determine which of these mappings are functions.

1. x is mapped to 2x - 1 where x is a real number

2. x is mapped to x^3 + 3 where x is a real number

3. x is mapped to 1/x - 1 where x is a real number

4. x is mapped to t where t^2 = x and x is a real number

5. x is mapped to the square root of x where x is a real number

6. x is mapped to the length of the line from the origin to (0, x) where x is a real number

Answer:

1. Yes
2. Yes
3. No, undefined when x = 0
4. No
5. Yes, x > or = 0
6. No, undefined when x < 0

Thank you for your kind offer of assistance, Dr. Peterson.

 

[Dr.Peterson]

I have attached the pictures of the pages for the questions and answers from the book.
As you can see, the question and its answer that I am struggling with is a part of a set of questions.
Having said that, this question is not related to previous questions or anything as you can see.

For your convenience, I copied the questions and answers from the book below - the same as the content of the attached pictures:

Question: Determine which of these mappings are functions.

1. x is mapped to 2x - 1 where x is a real number

2. x is mapped to x^3 + 3 where x is a real number

3. x is mapped to 1/x - 1 where x is a real number

4. x is mapped to t where t^2 = x and x is a real number

5. x is mapped to the square root of x where x is a real number

6. x is mapped to the length of the line from the origin to (0, x) where x is a real number

Answer:

1. Yes
2. Yes
3. No, undefined when x = 0
4. No
5. Yes, x > or = 0
6. No, undefined when x < 0

Thank you for your kind offer of assistance, Dr. Peterson.

There is one interesting thing in the context: these questions are given before "domain" has been defined for you! At this point, "function" is taken to mean "function defined on all real numbers". That is why they say "no" for #3, rather than saying "it would be a function if the domain were restricted to non-zero numbers". #4 is not a function for the usual reason, that it does not have a unique value for any input. #5 seems inconsistent with #3; here, too, the function is undefined for some values (x < 0), assuming you are not allowing complex outputs, but this time they say it is a function, but restrict its domain!

But you are right that this gives no new information about why they say #6 is undefined when x < 0.

The important thing is that the topic is functions, not lengths of lines, so you probably don't have to worry about this issue coming up again; we can just say that the book is wrong on that point (unless someone else sees an explanation), and continue working through the book's explanation of functions (which will give you more ways to talk about these problems when they have explained domain). Hopefully things will become clearer.

 

[Masaru]

Just have found another strange answer!

But you are right that this gives no new information about why they say #6 is undefined when x < 0.

The important thing is that the topic is functions, not lengths of lines, so you probably don't have to worry about this issue coming up again; we can just say that the book is wrong on that point (unless someone else sees an explanation), and continue working through the book's explanation of functions (which will give you more ways to talk about these problems when they have explained domain). Hopefully things will become clearer.

Yes, this book is confusing so it may be wrong on this point.

As you suggested, I will keep reading this book as I have not finished with the explanation about "domain" in it.

Will let you know if I find something important to understand this answer to the question provided in the book.

.......................................................................

Dr. Peterson & other people out there, I have just found another one that is beyond my comprehension...

This is not a part of or related to any other questions and the question is simply this:

If f (x) = the value of x correct to the nearest integer where x is a real number, find f (1.25), f (- 3.5), f (12.49).

And the answers provided in the text are:

1, not defined, 12

And I do not understand why f (- 3.5) is not - 4 because - 4 is an integer and we can round - 3.5 to - 4, which is the nearest integer to - 3 just like we can round +3.5 to +4, which is the nearest integer to 3.5.

I would much appreciate your help to understand this answer in the text.

​

 

[Dr.Peterson]

This is not a part of or related to any other questions and the question is simply this:

If f (x) = the value of x correct to the nearest integer where x is a real number, find f (1.25), f (- 3.5), f (12.49).

And the answers provided in the text are:

1, not defined, 12

And I do not understand why f (- 3.5) is not - 4 because - 4 is an integer and we can round - 3.5 to - 4, which is the nearest integer to - 3 just like we can round +3.5 to +4, which is the nearest integer to 3.5.

I would much appreciate your help to understand this answer in the text.

​

Probably what they are saying is that there are TWO "nearest integers" to -3.5, namely -3 and -4, which are the both 0.5 away from -3.5. So the function, taken exactly as stated, is not defined there.

In real life, we define "round to the nearest integer" in some way that disambiguates it and makes it a function; we may always "round up on 5", or "round away from zero on 5", or "round to even on 5". (You chose to round away from zero, following conventions you are most familiar with. The problem assumes that you take the definition literally and don't assume any convention.)

It's another somewhat confusing problem; I think they are just trying to come up with interesting examples using only simple functions, and accidentally running into trouble.

 

[Masaru]

I thought so, too, but this is certainly a bad question!

Probably what they are saying is that there are TWO "nearest integers" to -3.5, namely -3 and -4, which are the both 0.5 away from -3.5. So the function, taken exactly as stated, is not defined there.
.................................................................
It's another somewhat confusing problem; I think they are just trying to come up with interesting examples using only simple functions, and accidentally running into trouble.

Yes, initially I thought so, too, because -3.5 is exactly halfway between -3 & - 4.
So f (x) cannot be defined if the number x has " .5" on its end.

But this is such a bad tricky question...

By the way, I finished reading everything about domain & range in this text, but I could not get any information or clue or whatsoever that would help me to understand the answer to the previous question... So I still do not get why f(x) is not a function on the ground that it is undefined when x < 0 if x is mapped to the length of the line from the origin to (0, x) where x is a real number.

It is a mystery to me, and I have given up searching for a way to make sense of it...

 

[Dr.Peterson]

So I still do not get why f(x) is not a function on the ground that it is undefined when x < 0 if x is mapped to the length of the line from the origin to (0, x) where x is a real number.

It is a mystery to me, and I have given up searching for a way to make sense of it...

So have I.

I suppose you could write to the author or publisher; or you could tell me the name of the source (or the URL, since it looks like you are accessing it online) and I could see if there are any clues as to what they might mean or why they would give wrong answers ... but it doesn't seem worth the effort, since it's a side issue.

 

[Masaru]

The name of this book and publisher

So have I.

I suppose you could write to the author or publisher; or you could tell me the name of the source (or the URL, since it looks like you are accessing it online) and I could see if there are any clues as to what they might mean or why they would give wrong answers ... but it doesn't seem worth the effort, since it's a side issue.

Happy New Year To You, Dr. Peterson

The name of this book is:

Nelson Mathematics for Cambride International A Level Pure Mathematics 1

And the name of the publisher is:

Oxford University Press

As you suggested, I have just emailed them to make an enquiry about this question and answer that we cannot make sense of.

I will let you know if they send me a reply.

Thank you for your help, and I sincerely hope that you will have a really good year ahead!