#domain
tiny girderBOT
pale oxide
Chat am i doing this rightđ
looks like it, what are you trying to solve or do tho?
naive pivot
Chat am i doing this rightđ
the change you just made is correct yes but what youâre doing right or wrong necessarily cannot be determined unless you explain what youâre trying to solve
weak root
looks like it, what are you trying to solve or do tho?
im trying ot solve for the domain
weak root
the change you just made is correct yes but what youâre doing right or wrong nec...
im going into D:{x all real numbers / x=9 , x> 2/3 }
and my teacher likes to use number lines to help visually
but I do not udnerstand
acoustic anvil
if you're trying to find the domain of this function (I am assuming the expression on the LHS is the original function), I'd probably leave the function in its original form, and tackle the domain restriction caused by the x in the denominator first.
weak root
if you're trying to find the domain of this function (I am assuming the expressi...
Yes which I usually understand for equations such as
which I find pretty simple but my orginal equations for the one im struggling with is
acoustic anvil
oh, should have shown this first
weak root
omg mbđ
acoustic anvil
ok, turning it into the expression $\sqrt{\frac{3x+4}{x}}$ is not wrong and does not change the domain
fervent hawkBOT
Koyuki
acoustic anvil
so you're all good on that part
so you can probably work with this. though if you find the original expression easier to work with, we can proceed with that as well.
your pick.
weak root
No I feel like the expression would be easier to work with
acoustic anvil
which one
weak root
the second
weak root
new one
acoustic anvil
sure
weak root
yesyes
acoustic anvil
so consider the new one.
tackle the most obvious thing first - the fraction
what can't the denominator of a fraction be?
acoustic anvil
cool, and here the denominator is just x
so what's the first domain restriction we have?
weak root
x cantnot equal 0
acoustic anvil
cool, keep that in mind
now, for a square root to be defined, we need its argument to be non-negative (so $\geq 0$).
agreed?
fervent hawkBOT
Koyuki
weak root
Yes cannot be nonnegative
acoustic anvil
excellent
now, the sign of a fraction depends on the signs of the numerator and denominator.
if the two sides of the fraction have the same sign (+/+ or -/-) we get a positive result (or 0), and if they differ we get a negative result.
agreed?
weak root
yes
acoustic anvil
fantastic
now, we'll need to turn to a useful tool for determining the domains of rational functions - sign analysis
first, let's get the obvious out of the way.
when does the denominator change signs? (in other words, when is the denominator 0 for this case?)
weak root
When X isnt 0....?đ
acoustic anvil
so the denominator is 0 when x isn't 0?
careful - there's a reason I provided the second question to help you out
weak root
when X=0
acoustic anvil
cool!
so we know that the denominator is 0 when x = 0.
because the denominator is linear, we also know that when x > 0, the denominator is positive, and when x < 0, the denominator is negative.
agreed?
weak root
Yes
acoustic anvil
fantastic!
and just like the denominator, the numerator is linear.
so we know that for x < -4/3, the numerator is negative, and for x > -4/3, the numerator is positive.
agreed?
weak root
yes
acoustic anvil
I mean, is the numerator allowed to be 0?
weak root
yes..?
acoustic anvil
will it cause any problems if the numerator is 0?
acoustic anvil
yes..?
ok I'll take this answer
so the numerator is allowed to be 0, keep that in mind as well
and here is why we asked and answered all those questions.
weak root
ohhh so x is all real numberes expect for when X is less than 0 or greater than -4/3
acoustic anvil
now, the sign of a fraction depends on the signs of the numerator and denominato...
bringing back this concept, let's see the signs of the numerator and denominator for various intervals of x
acoustic anvil
ohhh so x is all real numberes expect for when X is less than 0 or greater than ...
oh yes I didn't see the except
yes, that is exactly the right concept!
acoustic anvil
bringing back this concept, let's see the signs of the numerator and denominator...
more rigorously, you would have done something like that
acoustic anvil
ohhh so x is all real numberes expect for when X is less than 0 or greater than ...
also btw, not or, but and
massive implications there
weak root
wait in which case would I use or
acoustic anvil
use or if the value should fulfill either inequality
but here, x must fulfill both inequalities for it to be excluded from the domain
weak root
Yes which I usually understand for equations such as
so this owuld be "or" correct?
acoustic anvil
that is fine with "or", yes
not your example
if you use your example, you'd find that what you are essentially saying is all real x, except all real x!
oh and by the way
acoustic anvil
ohhh so x is all real numberes expect for when X is less than 0 or greater than ...
"except for when x is less than 0" â
spot the mistake here
the greater than part is correct
weak root
its X is less or greater than 0? I thought that implied to greater than or equal to -4/3
acoustic anvil
nonono. remember what my very first question was?
I asked you what value the denominator cannot be
you mentioned x = 0
weak root
Ohhh yeah
acoustic anvil
which is correct, by the way
but that means you must include x = 0 as part of the exclusion
weak root
so in that case would both have the equal sign ?
weak root
but i thought the numerator could be 0
acoustic anvil
yeah the numerator can be 0
but you are writing the intervals of exclusion
not the intervals of inclusion
acoustic anvil
if you wrote the domain as intervals of inclusion then yes, there would be an equal sign on the x < -4/3, but not on the x > 0
weak root
now I see
X is all real numbers, X is greater than or equal to 0 and X is less than -4/3
acoustic anvil
uh
weak root
switching the greater and less
acoustic anvil
you put the equal sign on the wrong inequality if you are notating intervals of inclusion
x cannot be 0, remember?
weak root
wait is it possible to see a visual in the Domain
acoustic anvil
wdym
weak root
Yes which I usually understand for equations such as
something like the domain on the bottom
acoustic anvil
ok, so you want to notate the domain like in that image
yes, it is possible. you can try it now if you wish
weak root
idle raft