Functions problem

[shrewd python]

Can anyone help me solve this
We need to find range and domain of the function

[lucid birch]

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[rancid maple]

You're on the right track. Now, recall the range of arcsin(x).

[shrewd python]

minus pi/2 to pi/2

[rancid maple]

Right.

[rancid maple]

Well, the first inequality isn't quite right: -π/2 is excluded, since it would make the expression under the logarithm zero.

[shrewd python]

right.. so a normal paranthesis then

[rancid maple]

On which side?

[shrewd python]

on 10?

[-4, 10)

[rancid maple]

Yes.

[shrewd python]

like this?

[rancid maple]

Nice!

That's the domain.

[shrewd python]

this is done in my book also... but the thing im confused about is
we calculated the domain with ONLY the inequality given by range of arcsin....
we should consider that [pi + 2arcsin{ 3 - x /7 }]/pi greater than 0
because log wouldnt be defined in if we dont check that... we literally ignored that one

[rancid maple]

You literally started with that inequality, though...

[shrewd python]

yeah but later on we solved it thru that range of arcsin

[rancid maple]

Yes.

[shrewd python]

we didnt take intersection

[rancid maple]

Intersection of what?

[shrewd python]

of the 2 inequalities um

oh wait a minute

[rancid maple]

Oh, I see what you mean.
That isn't needed when dealing with a composite function.

As in, that happens "automatically", so to speak.

[shrewd python]

composite function?

[rancid maple]

Yeah, like here.

But if we had a sum or a product, then yes, we'd have to deal with them separately and intersect their domains.

[shrewd python]

whats that?

[rancid maple]

What's what?

[shrewd python]

composite function...

like

how can i directly assume log will always be defined

[rancid maple]

Oh.
Well, a composition of f(x) and g(x) is f(g(x)).

[rancid maple]

Because the inequality we took is how the domain of a logarithmic function is found.

[shrewd python]

okay so if g{x} is defined then f{x} will automatically defined?

[rancid maple]

Not sure what you mean.

We have a function of the form ln(f(x)). The condition to find its domain is just f(x) > 0.

[shrewd python]

did we do that? like we just directly put the range of f{x} and got that inequality of x...

[rancid maple]

Yes, that's the inequality you wrote.

[shrewd python]

does that mean if there is a function f{x} inside log
then we dont need to check for which values log will be defined and can directly find domain of the f{x}?

[rancid maple]

I'm not sure what exactly you're asking.
Logarithmic functions are defined when the expression under the logarithm is positive. That's it.

If we have a function of the form ln(f(x)), then you can just find its domain by solving f(x) > 0.

[shrewd python]

um... I started it fresh... and directly got the answer from range of arcsin...

I didn't even have to consider that entire thing being greater than 0

didn't even have to solve this

[shrewd python]

oh wait wait... this gives me the minimum value of arcsin for this function no?

like let that 3-x/7 = y
so arcsiny > -pi/2 .... {i}

and we know range is
-pi/2 <= arcsin y <= pi/2 ...... {ii}
taking intersection of both we get
-pi/2 < arcsin y <= pi/2 ....... {iii}
and solving {iii} gives us domain

[rancid maple]

If you don't do that, then you get the domain as [-4, 10], which is incorrect.

[shrewd python]

i think i get it..

lets say we have f[g{x}]
we can get domain of g{x} and then eliminate all values for f[g{x}] to be undefined... and finally we get the domain

[rancid maple]

Well, you can do that, yes. It's pretty much the same process, but not as straightforward.

[shrewd python]

right

thank you so much once again!

+close

[shy briar]

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[rancid maple]

You're welcome!

[shy briar]

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[rancid maple]

Remember to find the range, too.

[shrewd python]

yessir... its easy just will put range of arcsin
then multiply it with 2, followed by -1, inequalities flipped, add pi, then divide by pi. and then take log of the entire thing

will get something like minus infinity ig so will put paranthesis on that side