Question

If f(x) = radical(x) and f[g(x)] = 2 radical(x) , then g(x) =
Answer is 4x, but how...?

Answer 1

First, move forward to verify your answer, g(x)=4x.
Write f(x)=radical(x).
Replace the x in f(x) with g(x) to obtain f(g(x)). Replace the x in radical(x) with 4x.
Simplify the result. If you obtain 2radical(x), your choice of g(x) is correct (that is, 4x is correct).

Answer 2

How would I be able to figure it out without knowing the answer?

Answer 3

How would we move backwards? if f(g(x))=2 radical (x), re-write that '2' as radical (4).

Then f(g(x)) = 2 radical (x) = radical (4)*radical (x)=?

Answer 4

As y ou'll see above, I planned to and have taken you in both directions.

Answer 5

Can you combine radical(4) and radical(x) into just one radical?

Answer 6

So f(g(x)) = radical(4x)

Answer 7

\[\sqrt{4}*\sqrt{x}=\sqrt{?}\]

Answer 8

\[\sqrt{4x}\]

Answer 9

\[f(x) = \sqrt{x}\]

\[f(g(x))=2\sqrt{x}\]

thus

\[\sqrt{g(x)}=2\sqrt{x}\]

square both sides

Answer 10

Why did you put a radical around the left portion? @Zarkon

Answer 11

Albert: referring to your \[\sqrt{4x}\]... simplify this and then compare your result to the original problem statement. Do you see similarities or differences there?

Answer 12

I get \[2\sqrt{x}\]
That's the similarity that I see

Answer 13

Note that "radical(x)" in this context means sqrt(x).

Yes. Have you compared that to the original problem statement?

Answer 14

f[g(x)] = 2 radical(x)

Answer 15

Yes, so now I have to square both sides. But how shall I square the left portion?

Answer 16

Why do you have to square both sides? Zarkon has suggested that you do this, yes, but can you yourself see the reason for doing so?

Answer 17

Other than getting rid of the radical, then no, I do not see a reason for doing so

Answer 18

Getting rid of the radical is the whole idea. Then you're left with g(x) on the left side and 4x on the right side.

Answer 19

...this goes to show that g(x) is indeed = to 4x. g(x)=4x.

Answer 20

But how do we get rid of the f in f(g(x)). That's what I'm confused about now

Answer 21

I was confused at first also. Zarkon correctly pointed out that f(x) = sqrt(x).

So, he started with f(g(x)) and replaced the f( ) with sqrt( ).

Answer 22

so he / we ended up with Sqrt(g(x)) = 2Sqrt(x).
Square both sides of that. What do you obtain?

Answer 23

OH! Wow, this question has a bunch of twists and turns. Wow, I didn't even think of that. Now it all makes sense!

Answer 24

I'm so glad. Let's thank Zarkon.

Answer 25

We obtain 4x because in order to get rid of the radicals we have to raise both sides to the 2nd power.

Answer 26

Exactly.

Answer 27

Thank you @Zarkon! Thank you @mathmale!

Answer 28

:) :)