Let f(x)=√4x and g(x) = x + 6. What's the smallest number that is in the domain of fog?

- anonymous

Let f(x)=√4x and g(x) = x + 6. What's the smallest number that is in the domain of fog?

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- chestercat

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- jtvatsim

Well, at least we know how to find fog now. Check if your answer matches mine for the fog part.

- jtvatsim

\[f\circ g = f(g(x)) = \sqrt{4\cdot (x+6)}=\sqrt{4x + 24}\]

- arindameducationusc

sqrt(4)*(x+6)

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## More answers

- arindameducationusc

I think its not whole underoot

- anonymous

Ok we're getting somewhere.

- arindameducationusc

Ya @Generalman1230 , what is it?

- anonymous

Hold on, give me a sec guys.

- anonymous

Sorry, I just can't wrap it around my head. @jtvatsim @arindameducationusc

- anonymous

I thought I had something

- jtvatsim

No worries, I think we are majoring on minors right now. The question that arindameducationusc doesn't have to do with the process I taught you. It's just a matter of "before we get carried away" let's check that we actually wrote the equation for f correctly.

- arindameducationusc

@Generalman1230 ,
@jtvatsim will take care.... don't worry.....
:)

- jtvatsim

OK, let's try this from the top again. I think you are fine. :)

- jtvatsim

We have \[f(x) = \sqrt{4x}\] where the 4x is completely inside the root, and \[g(x) = x+6\].

- jtvatsim

We then know based on our previous conversation that \[f \circ g = f(g(x)) = f(x+6) = \sqrt{4 \cdot (x+6)} = \sqrt{4x + 24}\]

- jtvatsim

That's where we left off before. Are you OK up to that step?

- anonymous

Yes, I am.

- jtvatsim

Excellent. Now, the question wants us to find the smallest number in the domain. How small can I get before there is a problem? Well, let's experiment! Time for science!

- jtvatsim

Or even better, we can stop and think about what can and can't happen.

- jtvatsim

The only issue in this situation (there are no fractions, so dividing by 0 isn't a problem) is if we get a negative number inside the square root.

- jtvatsim

The smallest thing that can EVER be inside a square root is 0 then. It's the first "not-negative" number. Does that logic make sense so far?

- jtvatsim

Phew... got caught in an internet black hole there, don't know what happened.

- anonymous

The website crashed, I guess. I couldn't get on it for a second.

- jtvatsim

Yep, it's still a little shaky even now...

- jtvatsim

So, are you comfortable with the idea that the smallest thing you can have inside a square root is 0? There is one more step to do after coming to grips with this fact.

- anonymous

I tried to type in yes, but the site crashed again.

- jtvatsim

lol, almost there site, just hang in there.

- jtvatsim

K, so I'll make this quick before it crashes again.

- jtvatsim

The smallest thing that can be inside is 0. BUT... we have a HUGE equation inside right now.
That's not a problem, the smallest this HUGE equation inside can be is 0, so... this means:
4x + 24 = 0 is the smallest we can go,
4x = -24 is the smallest we can go,
x = -6 is the smallest we can go.
x = -6 is the smallest number in the domain.

- anonymous

@jtvatsim I can't thank you enough for all the help you gave throughout the past hour. I know you have to get off but could you help me with one more problem?

- anonymous

If you have to go now, then that's totally understandable.

- jtvatsim

Gotcha, you know I do have to run. But message me the question, and I can message you back sometime tomorrow. If it's due before then, there are plenty of other great tutors online. Good luck! You have come far! :)

- anonymous

@jtvatsim Will do, thanks again.

- jtvatsim

You are very welcome! Summit Mt. Math and conquer it! >:) Nights!

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