Using Domain Convention

[Probability]

Could somebody please explain how to use intervals in conjunction with functions.

g(x) = 1/x

If 1/0 is not allowed, then how does this work?

h(x) = 1/x

The function h has a domain R excluding 0, which consists of two open intervals (- infinity, 0) and (0, infinity)

It's an interesting subject but until somebody explains how intervals are related to functions I cant move forward with this subject as the text I have does not explain how to use it?

 

[Mrspi]

Could somebody please explain how to use intervals in conjunction with functions.

g(x) = 1/x

If 1/0 is not allowed, then how does this work?

h(x) = 1/x

The function h has a domain R excluding 0, which consists of two open intervals (- infinity, 0) and (0, infinity)

It's an interesting subject but until somebody explains how intervals are related to functions I cant move forward with this subject as the text I have does not explain how to use it?

I wonder if you are making MORE of the topic of "intervals" than it deserves.

In my opinion, interval notation is simply a shorthand way of indicating the elements of a set.

In your example, f(x) = 1/x, the domain is the set of all numbers which can be used in place of x. Since 1/0 is NOT DEFINED, we surely cannot use 0 as a value of x. Any other value of x is acceptable. So, we could say in words, "x can be any real number except 0"

Or, we can show the same thing in interval notation: x is an element of {(-infinity, 0) U (0, infinity)}.

Or....x is a real number such that x < 0 OR x > 0

Or....x is a real number and x <>0.

All of those statements describe the same set of numbers.

If I have misunderstood your question, please repost with a more specific question.

 

[Probability]

I wonder if you are making MORE of the topic of "intervals" than it deserves.

In my opinion, interval notation is simply a shorthand way of indicating the elements of a set.

In your example, f(x) = 1/x, the domain is the set of all numbers which can be used in place of x. Since 1/0 is NOT DEFINED, we surely cannot use 0 as a value of x. Any other value of x is acceptable. So, we could say in words, "x can be any real number except 0"

Or, we can show the same thing in interval notation: x is an element of {(-infinity, 0) U (0, infinity)}.

Or....x is a real number such that x < 0 OR x > 0

Or....x is a real number and x <>0.

All of those statements describe the same set of numbers.

If I have misunderstood your question, please repost with a more specific question.

Thank you for the explanation, I understand what you are writing and think a lot of this problem is down to my in-experience in the subject itself.

I understand that "x" can be any value, it is a variable, but my main misunderstanding is the "interval notation", I just can't seem to grasp the understanding of how it "converts over" if that is worded sensibly?

Keeping with the above example; h(x) = 1/x

The domain (x) in this example is a real number R excluding 0, which I am then told consists of two open intervals (- infinity, 0) and (0, infinity)

Now the misunderstanding?

In this example above, the domain (x) can be any real number "negative or positive", so I think ( - infinity) or (infinity) is probably referring to the "x", but because the example clearly states "0" is excluded, then how does (- infinity, 0) and (0, infinity) the zeros in the parentheses become included?

 

[Deleted member 4993]

By convention, if we say,

domain of function is (a,b) then a and b are excluded from the domain - the parentheses are defined to say that.

But if we say,

domain of function is [a,b] then a and b are included in the domain - the box-brackets are defined to say that.

 

[mmm4444bot]

the domain (x) can be any real number

I'm not certain of your situation, but perhaps you are misunderstanding the definition of "domain".

Do not use the symbol x to represent a domain, as you seem to have done above.

Also, it is not correct to state that some domain "can be any real number".

The domain is a set of numbers; the symbol x represents only one of these numbers at a time. (Sometimes, we know or we search for a particular value of x; other times, we need to assume that x could represent any number in the domain.)

It is because the value of x can vary that we call it a "variable" symbol.

If we were to list all of the possible values that the symbol x could take on for some function, then that list comprises the function's domain.

Again, the domain is the set of all possible values of x that a function accepts as input.

Many functions accept an infinite number of values. In all of these cases, nobody can possibly list all of the values; hence, we need to express the "list" using some sort of set notation, like "interval notation".

If you're not sure about the difference between the concepts of "x" or "variable" or "domain", then that uncertainty is probably why you're confused. If you're confused about something else, please be specific.

Cheers ~ Mark :cool:

 

[Probability]

I'm not certain of your situation, but perhaps you are misunderstanding the definition of "domain".

Do not use the symbol x to represent a domain, as you seem to have done above.

Also, it is not correct to state that some domain "can be any real number".

The domain is a set of numbers; the symbol x represents only one of these numbers at a time. (Sometimes, we know or we search for a particular value of x; other times, we need to assume that x could represent any number in the domain.)

It is because the value of x can vary that we call it a "variable" symbol.

If we were to list all of the possible values that the symbol x could take on for some function, then that list comprises the function's domain.

Again, the domain is the set of all possible values of x that a function accepts as input.

Many functions accept an infinite number of values. In all of these cases, nobody can possibly list all of the values; hence, we need to express the "list" using some sort of set notation, like "interval notation".

If you're not sure about the difference between the concepts of "x" or "variable" or "domain", then that uncertainty is probably why you're confused. If you're confused about something else, please be specific.

Cheers ~ Mark :cool:

Thanks for your reply Mark

According to my coursebook the function of "x", "x" is a variable and has been named the domain, also I am to understand that the domian can be presented like (-1 < x < 1).

The problem I have is converting the algebra to interval notation, so the inequality above would be a closed interval, then if I had to write that as interval notation it would be {a, b} "These brackets representing square brackets". This is direct from my coursebook.

but if I were given an example like as previously posted, i.e. h(x) = 1/x, and I am asked to write this as interval notation, then this is were my problem is becasue I don't know how to "change it" to "interval notation" of (- infinity , 0)(0, infinity)

Thanks

 

[mmm4444bot]

According to my coursebook … "x" is a variable and has been named the domain

Your coursebook is wrong. The symbol x is never named "the domain".

The word "domain" is what we use when we want to refer to the entire set of numbers which contains all possible values that x can represent (for some given function, of course).

I am to understand that the domian can be presented like (-1 < x < 1).

The notation above is wrong. Your coursebook should not use parentheses around a compound inequality. The book should display the expression simply as -1 < x < 1.

Your book seems to be mixing two different systems of notation together; that's a bad idea.

The problem I have is converting [inequalities or set notation] to interval notation

Did your coursebook previously introduce the concept of the Real number line?

Have you yet graphed sets of numbers on the Real number line?

You seem to be missing something basic and to be operating under some misconceptions. If you accurately quoted your coursebook above, then you're using a lousy coursebook (i.e., too many errors), and you need to find another.

When discussing math in writing, we all need to pay attention to our English. Using poor English makes matters worse because mathematics is very unforgiving of sloppy communication. Math needs to be explained clearly.

In addition to looking for a qualified coursebook, I think that you would also benefit from a few sessions face-to-face with a qualified tutor.

Cheers :cool:

 

[Probability]

Your coursebook is wrong. The symbol x is never named "the domain".

The word "domain" is what we use when we want to refer to the entire set of numbers which contains all possible values that x can represent (for some given function, of course).




The notation above is wrong. Your coursebook should not use parentheses around a compound inequality. The book should display the expression simply as -1 < x < 1.

Your book seems to be mixing two different systems of notation together; that's a bad idea.

Did your coursebook previously introduce the concept of the Real number line?

Have you yet graphed sets of numbers on the Real number line?

You seem to be missing something basic and to be operating under some misconceptions. If you accurately quoted your coursebook above, then you're using a lousy coursebook (i.e., too many errors), and you need to find another.

When discussing math in writing, we all need to pay attention to our English. Using poor English makes matters worse because mathematics is very unforgiving of sloppy communication. Math needs to be explained clearly.

In addition to looking for a qualified coursebook, I think that you would also benefit from a few sessions face-to-face with a qualified tutor.

Cheers :cool:

The exact wording in the book says; In general, a function is specified by giving; A set of allowed input values, called the domain of the function. The book then shows a oval with an "x" inside with an arrow pointing towards a box called process squaring, this the book says is the rule of the function, for converting each input value into a unique output value.

The book goes on to say that the function f uses the notation f(x) = ... to specify a rule of f, so y = f(x).

In the above regarding the oval and "x", the book writes f(x) = x^2.

Then the book goes on to say various notations can be used to specify both the rule of a function and its domain, example;


f(x) = x^2 + 1 (0< x < 6)

The book says the function f with rule f(x) = x^2 + 1 whose domain is the set of real numbers x satisfies

0 < x < 6 { Are these intervals or domain }

Note that brackets are now removed for some reason?

Note an example question from the book.

Describe the domain of the function f given by the specification

f(t) = ( -1 < t < 2) Notice the use of brackets.

answer

The domian of f is the set of real numbers t satisfying - 1 < t < 2. Notice no mention of brackets?

For me the book seems to be written back to front, it talks about function and domian but does not explain for me clearly, then the book asks questions about it, then gives worked solutions and the solutions include ideas the book has not introduced until the next few pages along?

 

[Mrspi]

The exact wording in the book says; In general, a function is specified by giving; A set of allowed input values, called the domain of the function. The book then shows a oval with an "x" inside with an arrow pointing towards a box called process squaring, this the book says is the rule of the function, for converting each input value into a unique output value.

The book goes on to say that the function f uses the notation f(x) = ... to specify a rule of f, so y = f(x).

In the above regarding the oval and "x", the book writes f(x) = x^2.

Then the book goes on to say various notations can be used to specify both the rule of a function and its domain, example;


f(x) = x^2 + 1 (0< x < 6)

The book says the function f with rule f(x) = x^2 + 1 whose domain is the set of real numbers x satisfies

0 < x < 6 { Are these intervals or domain }

Note that brackets are now removed for some reason?

Note an example question from the book.

Describe the domain of the function f given by the specification

f(t) = ( -1 < t < 2) Notice the use of brackets.

answer

The domian of f is the set of real numbers t satisfying - 1 < t < 2. Notice no mention of brackets?

For me the book seems to be written back to front, it talks about function and domian but does not explain for me clearly, then the book asks questions about it, then gives worked solutions and the solutions include ideas the book has not introduced until the next few pages along?

Ok...what we have here (I think!) is a "failure to communicate."

When your book says this:

f(x) = x^2 + 1 (0< x < 6)

The parentheses are NOT INTENDED to be part of the domain description. They are intended to "set off" the domain conditions from the rest of the statement.

Here's what that statement means:

f(x) = x^2 + 1 WHERE 0 < x < 6

The "rule" describing the function is "x^2 + 1", the NAME of the function is f(x), and the set of input values is the set of real numbers beginning with 0 and ending with 6.

The DOMAIN of the function, described as a set, is the set of all real numbers which are greater than or equal to 0, and less than or equal to 6. Since both 0 and 6 are part of the domain, we can use interval notation to write the domain this way: [0, 6]. The square brackets indicate that 0 and 6 are INCLUDED.

If our interval was 0 < x < 6, 0 and 6 are NOT included and we'd show that as (0, 6).


If I tell you that g(x) = 3x - 5 where -2 < x < 7, then the set of numbers we are allowed to use for x are the real numbers which are greater than -2, and are less than or equal to 7. As an interval, that domain would be (-2, 7]. Notice that we have a parentheses on the left end, indicating that -2 is NOT part of the interval. We have a square bracket on the right end, indicating that 7 IS part of the interval.

I guess this is my last and final shot at attempting to explain this. I'm thinking that about 5 minutes with your teacher, where YOU ask "What does this mean here when it says 'f(x) = x^2 + 1 (0< x < 6)' would have cleared this whole thing up......

 

[Probability]

Ok...what we have here (I think!) is a "failure to communicate."

When your book says this:

f(x) = x^2 + 1 (0< x < 6)

The parentheses are NOT INTENDED to be part of the domain description. They are intended to "set off" the domain conditions from the rest of the statement.

Here's what that statement means:

f(x) = x^2 + 1 WHERE 0 < x < 6

The "rule" describing the function is "x^2 + 1", the NAME of the function is f(x), and the set of input values is the set of real numbers beginning with 0 and ending with 6.

The DOMAIN of the function, described as a set, is the set of all real numbers which are greater than or equal to 0, and less than or equal to 6. Since both 0 and 6 are part of the domain, we can use interval notation to write the domain this way: [0, 6]. The square brackets indicate that 0 and 6 are INCLUDED.

If our interval was 0 < x < 6, 0 and 6 are NOT included and we'd show that as (0, 6).


If I tell you that g(x) = 3x - 5 where -2 < x < 7, then the set of numbers we are allowed to use for x are the real numbers which are greater than -2, and are less than or equal to 7. As an interval, that domain would be (-2, 7]. Notice that we have a parentheses on the left end, indicating that -2 is NOT part of the interval. We have a square bracket on the right end, indicating that 7 IS part of the interval.

I guess this is my last and final shot at attempting to explain this. I'm thinking that about 5 minutes with your teacher, where YOU ask "What does this mean here when it says 'f(x) = x^2 + 1 (0< x < 6)' would have cleared this whole thing up......

Thank you, your explantion is perfect, now I understand it:smile: