Thread: Find a function

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

Originally Posted by Zeppelin

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

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

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

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.

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

so taking k-th roots:

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

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

RonL