Question

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

Answer 1

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

Answer 2

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

Answer 3

sqrt(4)*(x+6)

Answer 4

I think its not whole underoot

Answer 5

Ok we're getting somewhere.

Answer 6

Ya @Generalman1230 , what is it?

Answer 7

Hold on, give me a sec guys.

Answer 8

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

Answer 9

I thought I had something

Answer 10

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.

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

Yes, I am.

Answer 17

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!

Answer 18

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

Answer 19

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.

Answer 20

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?

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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.

Answer 25

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

Answer 26

lol, almost there site, just hang in there.

Answer 27

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

Answer 28

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.

Answer 29

@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?

Answer 30

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

Answer 31

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! :)

Answer 32

@jtvatsim Will do, thanks again.

Answer 33

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