# Let f(x)=x^2 and g(x)=sqrtx, how do you find the domain and rules of (f*g)(x)?

Jan 31, 2017

$f \left(g \left(x\right)\right) = x , x \ge 0$.

#### Explanation:

Seeing as this is posted under function composition, I'll treat

$\left(f \cdot g\right) \left(x\right)$ as the composition $f \left(g \left(x\right)\right)$.

The domain of $f$ is $\mathbb{R}$, since any real $x$ can be squared.

The domain of $g$ is $\left[0 , + \infty\right)$ because roots only accept non negative input.

The composition is defined, when $x$ is in the domain of $g$, and $g \left(x\right)$ is in the domain of $f$. In other words:

With $f \left(g \left(x\right)\right)$, $x$ is the input (independent variable) for $g$, and $g \left(x\right)$ is the input for $f$. Therefore, the inputs each must be part of the domains of the functions they are to be used in. We have the following constraints:

$x$ must be non-negative, since it goes into the function $g \left(x\right) = \sqrt{x}$.

$g \left(x\right)$ must be real (it obviously is).

Therefore,

$f \left(g \left(x\right)\right) = {\left(\sqrt{x}\right)}^{2} = x$, for $x \ge 0$.