Domain search for logarithmic function: log_x(2x), f(x) = log_2(sin x/4x+2)

[e_2.718281828]

I'm so sorry for the terrible technical english, I go to a school where we speak french, so the words you use for maths will be different, but I will try to explain them as I can. Please correct any english/math syntax mistakes I make. I hope you can help me figure out if my answer is right or wrong.



I'm looking for the domain of this function. I believe domain in english is range but i'm not sure so i'll give an example :
if g(x) = logx 2x then dom g = [0; -> \ {1}
(read as "log base x of (2x))

back to my problem
for f(x) = log2(sin x/4x+2)
(read it as "log base 2 of (sin x/(4x+2))" )

f(x) can only exist if (sin x/(4x+2)) is strictly greater than 0 and if (4x+2) is different than 0
so x has to be different than -1/2
to be sure that (sin x/4x+2) is strictly greater than 0, (sin x) and (4x+2) must both be POSITIVE or both be NEGATIVE
I wrote what we call in french a "sign table" picture :

with this "sign table" I can see what the domain should look like, but sin x is a periodic function, so the domain gets VERY messy.

the answer I got is
dom f=for every whole, negative and non-zero variable called "k" ]2kPi - Pi;2kPi[ then ]-Pi;-1/2[ then for every natural number "l" ]2lPi ; Pi+2lPi[

 

[JeffM]

I'm so sorry for the terrible technical english, I go to a school where we speak french, so the words you use for maths will be different, but I will try to explain them as I can. Please correct any english/math syntax mistakes I make. I hope you can help me figure out if my answer is right or wrong.



I'm looking for the domain of this function. I believe domain in english is range but i'm not sure so i'll give an example :
if g(x) = logx 2x then dom g = [0; -> \ {1}
(read as "log base x of (2x))

back to my problem
for f(x) = log2(sin x/4x+2)
(read it as "log base 2 of (sin x/4x+2)" )

f(x) can only exist if (sin x/4x+2) is strictly greater than 0 and if (4x+2) is different than 0
so x has to be different than -1/2
to be sure that (sin x/4x+2) is strictly greater than 0, (sin x) and (4x+2) must both be POSITIVE or both be NEGATIVE
I wrote what we call in french a "sign table" picture :

with this "sign table" I can see what the domain should look like, but sin x is a periodic function, so the domain gets VERY messy.

the answer I got is
dom f=for every whole, negative and non-zero variable called "k" ]2kPi - Pi;2kPi[ then ]-Pi;-1/2[ then for every natural number "l" ]2lPi ; Pi+2lPi[

In English, the "domain" of a single valued function is the set of numbers that the independent variable may be drawn from, and the "range" of a function is the set of numbers that the function may attain.

The domain of g(x) given \(\displaystyle g(x) = log_x(2x)\) is \(\displaystyle (0,\ 1) \ \bigcup \ (1,\ \infty).\)

All positive numbers except one are permitted as x.

The range of g(x) is \(\displaystyle (-\ \infty,\ \infty).\)

Now I take it that

\(\displaystyle f(x) = log_2 \left ( \dfrac{sin(x)}{4x + 2} \right ).\)

That is not what you wrote, but it appears to be what you are talking about.

 

[Dr.Peterson]

back to my problem
for f(x) = log2(sin x/4x+2)
(read it as "log base 2 of (sin x/4x+2)" )

f(x) can only exist if (sin x/4x+2) is strictly greater than 0 and if (4x+2) is different than 0
so x has to be different than -1/2
to be sure that (sin x/4x+2) is strictly greater than 0, (sin x) and (4x+2) must both be POSITIVE or both be NEGATIVE
I wrote what we call in french a "sign table" picture :

View attachment 9140

with this "sign table" I can see what the domain should look like, but sin x is a periodic function, so the domain gets VERY messy.

the answer I got is
dom f=for every whole, negative and non-zero variable called "k" ]2kPi - Pi;2kPi[ then ]-Pi;-1/2[ then for every natural number "l" ]2lPi ; Pi+2lPi[

Your work looks excellent; I, too, might call that a "sign table". (There is really no one standard name for it, and there are several ways it can be written.)

I will just make a few comments JeffM didn't make:

The function must be written as f(x) = log_2(sin(x)/(4x+2)), in order to mean what you intend. In typed form, the underscore is needed to show that the 2 is a subscript, and the parentheses around the denominator are required. Parentheses around x are recommended. Your notation in the written form is clearer than what you typed (mostly the word "then" being confusing), but you have done well.

Your notation for intervals is standard in France, and perhaps elsewhere; in the English-speaking world I think your ]2kPi - Pi;2kPi[ would usually be (2kPi - Pi, 2kPi), and I myself would write ((2k - 1)pi, 2k pi) to emphasize that it is from an odd multiple of pi to an even multiple of pi. That is a matter of taste.

The one error in your answer is that in the last part you take "l" to be a natural number (which to me is a positive integer - not zero), so that you have excluded ]0;Pi[. Does "natural" for you include zero?

 

[e_2.718281828]

Your work looks excellent; I, too, might call that a "sign table". (There is really no one standard name for it, and there are several ways it can be written.)

I will just make a few comments JeffM didn't make:

The function must be written as f(x) = log_2(sin(x)/(4x+2)), in order to mean what you intend. In typed form, the underscore is needed to show that the 2 is a subscript, and the parentheses around the denominator are required. Parentheses around x are recommended. Your notation in the written form is clearer than what you typed (mostly the word "then" being confusing), but you have done well.

Your notation for intervals is standard in France, and perhaps elsewhere; in the English-speaking world I think your ]2kPi - Pi;2kPi[ would usually be (2kPi - Pi, 2kPi), and I myself would write ((2k - 1)pi, 2k pi) to emphasize that it is from an odd multiple of pi to an even multiple of pi. That is a matter of taste.

The one error in your answer is that in the last part you take "l" to be a natural number (which to me is a positive integer - not zero), so that you have excluded ]0;Pi[. Does "natural" for you include zero?

thanks for the corrections!
yes natural numbers are every positive whole number and include 0
I use ]2Pi ; 0[ to say that 2Pi and 0 are NOT in the domain of f(x)
»l » can be 0 because ]0;Pi[ is a part of the domain? I think that’s how it works at least. Thanks for the imput, This is actually a decent forum and I look forward to use it more often!

 

[e_2.718281828]

In English, the "domain" of a single valued function is the set of numbers that the independent variable may be drawn from, and the "range" of a function is the set of numbers that the function may attain.

The domain of g(x) given \(\displaystyle g(x) = log_x(2x)\) is \(\displaystyle (0,\ 1) \ \bigcup \ (1,\ \infty).\)

All positive numbers except one are permitted as x.

The range of g(x) is \(\displaystyle (-\ \infty,\ \infty).\)

Now I take it that

\(\displaystyle f(x) = log_2 \left ( \dfrac{sin(x)}{4x + 2} \right ).\)

That is not what you wrote, but it appears to be what you are talking about.

I used a backslash (\) to say that 1 does not belon to the domain, and an arrow to the right (->) to mark +infinity. Thanks for the correction on the parenthesis, I understand the possible confusion