Is intuition able to provide an answer to this, or is there another way?

[weary mesa]

You can assume the domain is always R unless you have restrictions like division by 0, negative number under the square root or non-positive numbers in the logarithm.

The range is more complicated, it requires analyzing the function in terms of extrema, limits, monotony etc.

[flint tapir]

More than half of these terms are unfamiliar to me, but it's okay.

[weary mesa]

My bad haha

extrema are maxima or minima

[flint tapir]

I did see something in the internet where they tested 2, 1, 0, 1, 1, and 2 to know the domain

[weary mesa]

limits I meant like horizontal/vertical asymptotes

[flint tapir]

Should I learn that first? or maybe in spare time?

[weary mesa]

monotony refers to the behavior of a function in terms of wheter it's incresing or decreasing

,w plot y = x^2

[raw panther]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by adoniszgks

[weary mesa]

this functions has a minimum

at x = 0

and it's increasing for x > 0

so from that you can conclude its range [0,oo)

[flint tapir]

increasing on the right side? and decreasing on the left?

[weary mesa]

yess

and it only decreases until that minimum

[flint tapir]

I see.

[weary mesa]

then it increases

[weary mesa]

so here

i should demand that you know basic functions

like x²

x

sqrt(x)

1/x

and like that

[weary mesa]

because from that you can conclude similarities

[flint tapir]

I haven't read the entire function part yet (really takes time to learn new stuff)

But I think that number 2 cannot be 0 or negative so the domain is [1, inf]

[flint tapir]

What do you mean by similarities, by the way? Is that a topic or the actual definition of it?

[weary mesa]

i came up with that

[weary mesa]

not quiet

,, y = \sqrt{5x+10}

[raw panther]

𝔸dωn𝓲²s

[weary mesa]

if x = 0

I get sqrt(10)

still positive inside

So instead what you are looking out for is

5x + 10 < 0

and exclude these solutions from the domain

[flint tapir]

ohh

[weary mesa]

alternatively you can consider

[raw panther]

𝔸dωn𝓲²s

[weary mesa]

these solutions tell what x are in the domain

only them would be allowed

so if you solve it

you get

either

[raw panther]

𝔸dωn𝓲²s

[weary mesa]

try 2. or 4.

in 3.

you can also consider the argument (what's inside the square root)

y = 5x+10 is a linear functions

,w plot y = 5x+10

[raw panther]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by adoniszgks

[weary mesa]

it's negative if x < -2

you can also see it that way

if you try 4. on your own now

it's easier to consider the argument graphically

[flint tapir]

Oh my bad, I was doing something, ima solve that later, so sorry.

[flint tapir]

Okay, I've been rereading it for awhile, and I am a bit confused about starting this bad (sorry I was doing chores).

[weary mesa]

no sorry just show your work

or what is it

[flint tapir]

What you are trying to ask is to substitute 2.4 here? (sorry I am lost in that part)

[weary mesa]

no

[flint tapir]

Sorry, I don't understand what you're trying to imply here. Either I am too dumb to understand or I am being dyslexic

[weary mesa]

You can either calculate which x are illegal (1st) or you can calculate which are in the domain (allowed) (2nd)

Do you see how they complement each other

x < 2 is the complement of x >= 2

or basically "opposite"

Either we say R{x < -2}

Or x >= -2

[flint tapir]

So anything less than -2 is illegal, and anything greater than or equal to -2 is allowed. For what I understand, these are two ways to find the domain of the function.

From what I understand, they are both trying to imply the same thing.

-2 < x < inf

or

[-2,inf)?

[weary mesa]

yes

Except it's actually

-2 <= x < oo

-2 allowded too

Makes the inner part 0

so sqrt(0) if x = -2

Which is still valid

[flint tapir]

my bad, I forgot the equal

[weary mesa]

yeah np

You got the idea

Thats important

[flint tapir]

So since anything below 0 is not part of the domain,
5x + 10 < - 1 (illegal)
5x + 10 >= 0 (legal)

therefore

[0, inf)

[weary mesa]

No

5x + 10 < 0

and we just said it's [-2, inf)

[flint tapir]

dammit, I am not usning my brain right, I FORGOT ABOUT THE MULTIPLICATION MY BAD SO SORRY

[weary mesa]

haha what

[flint tapir]

I forgot about 5(x)

[weary mesa]

Here a quick sheet

[flint tapir]

5(-2) + 10 = 0 (I need to get used to these stuff)

[raw panther]

𝔸dωn𝓲²s

[flint tapir]

So the trick here is to focus on one area where it can cause an error, such as the square root or denominator in a fraction.

[weary mesa]

hope you can start something with this

[weary mesa]

Yes

Because there it isnt defined

[flint tapir]

@weary mesa By the way, thank you so much for your time. I still have a very long road to actually reach mathematical maturity, but I am trying everyday.

[weary mesa]

I disagree

there is no mathematical maturity imo

[flint tapir]

??

[weary mesa]

The more you get into math

The more inmature things get

Like a toxic relationship

[flint tapir]

I kept hearing that term, and I imagine it as some sort of having a greater understanding of the concept.

[weary mesa]

yea 🥱

[flint tapir]

lol

aight, I think we're good here. thank you

.close

[jovial dawn]

Solved

Post marked as solved by @dusky kayak.

Use .unsolved if this was a mistake.

[flint tapir]

I have to take note of this stuff, especially these techniques, so that I can easily review it.

[weary mesa]

I did it for you

This is basically what should be in your mind

[weary mesa]

This here

[flint tapir]

understood

[weary mesa]

good

Now do 4

[flint tapir]

?

[weary mesa]

task 4 bro

[flint tapir]

oh right, my bad

x^2 - 3x >= 0

so to make this 0 then 3, since 3^2 = 9 and 3*3 = 9

[3, inf)

So anything below 3 is illegal (result in negative)

for example

2^2 - 3(2)

4 - 6 = -2 which is below 0

[weary mesa]

,w plot y=x^2-3x

Not quiet

[raw panther]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by adoniszgks

[weary mesa]

What about the other side

x < 0?

You forgot one

[flint tapir]

but is the other one correct?

you mean the allowed one?

[weary mesa]

Yes

It's part of the solution

[flint tapir]

Oh, so the solution consist of illegals and legal?

[weary mesa]

Thats why I said you consider it graphically too

y = x² - 3x = x(x-3)

Roots x = 0 or x = 3

The parabola is open upwards

Thus

(-oo, 0] U [3,oo)

For this interval

x²-3x is positive or 0

[flint tapir]

Though it is already midnight and I have to go to bed, I think I have trouble looking it up in graphs. If that is okay with you, I will come back here in the morning.

@weary mesa btw thanks once again

[weary mesa]

yeaa

so sentimental

[flint tapir]

By the way aren't we doing like a square root? so anything below 0 is not part of domain?

Also the black dot is the negative inf to 0 then the red is 3 to inf?

Am I looking at it in the right way?

[flint tapir]

@weary mesa I forgot to tag you lol, this might be hidden on discord

[weary mesa]

I dont understand the first question. The second statement ist correct.

[flint tapir]

I mean, x^2 - 3x came from question 4, which is square rooting x^2 - 3x. Am I looking at it in the right way? I am confused. I thought negative cannot be square-rooted, or is my algebra wrong?

Oh nvm, squared is more than -3(n)

even with negatives such as x, -3 will make it positive, my bad