Question

Express the following function, F(x) as a composition of two functions f and g.
F(x) = x^2/(x^2+4)
@wio

Answer 1

Well, the easy way is to say \(g(x) =1\) and \(f(x) = F(x)\).

Answer 2

ok

Answer 3

Another way is to say \[
F(x) = \frac{x^2}{x^2+4} = \frac{f(x)}{g(x)}
\]Which means \[
f(x)=x^2\\
g(x) = x^2+4\\
F =\frac fg
\]

Answer 4

so i just substitute?

Answer 5

I guess so

Answer 6

ok

Answer 7

do you know how to do it? can you show me?

Answer 8

@wio

Answer 9

I think I already showed you how to do it.

Answer 10

Wait, actually

Answer 11

Okay, I messed up, let me do it again.

Answer 12

Another way is to say \[
F(x) = \frac{x^2}{x^2+4} = \frac{(x^2)}{(x^2)-4}
\]If we let \(g(x) = x^2\): \[
F(x)=\frac{g(x)}{g(x)+4}
\]Now, we want \[
f(g) = \frac{g}{g+4}
\]Which means that \[
f(x) =\frac{x}{x+4}
\]So now we are done because \[
F(x) = (f\circ g)(x)
\]

Answer 13

So now, my final answer is \[x / x + 4\]

Answer 14

That is just \(f(x)\).
Your final answer is providing \(f(x)\) and \(g(x)\).

Answer 15

ok so how do I get g(x)? We found f(x) already right?

Answer 16

We already did get \(g(x)\), just look up.

Answer 17

oh, okay. thanks!

Answer 18

x/ x + 4 AND g/ g + 4 ARE MY FINAL ANSWERS RIGHT? @wio