Find a function

**Zeppelin** · Dec 2nd 2006, 01:36 AM

Find a function $f(x)=x^k$ and a function $g$ such that $f(g(x))=h(x)=\sqrt{3x+x^2}$ .

**CaptainBlack** · Dec 2nd 2006, 01:49 AM

Originally Posted by Zeppelin

Find a function $f(x)=x^k$ and a function $g$ such that $f(g(x))=h(x)=\sqrt{3x+x^2}$ .

The wording here is a bit confused, but I will assume that we seek a function $g(x)$ , such that if $f(x)=x^k$ then:

$f(g(x))=h(x)=\sqrt{3x+x^2}$

For the moment I will not worry about the domain/s of the functions but go through a formal manipulation of the expressions and then worry about the domains later.

$f(g(x))=(g(x))^k=\sqrt{3x+x^2}=(3x+x^2)^{1/2}$ ,

so taking k-th roots:

$g(x)=(3x+x^2)^{1/2k}$ ,

and the domain of $g$ and $h$ is $\{x: x \in \mathbb{R}, \mbox{ and }x \le -3,\ \mbox{ or } x \ge 0\}$

RonL

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