# How do you find the domain of f(g(x)) when f(x) = 1/(x+3) and g = (-2/x)?

Aug 3, 2017

The domain is ${D}_{f} \left(g \left(x\right)\right) = \mathbb{R} - \left\{\frac{2}{3}\right\}$

#### Explanation:

We have,

$f \left(x\right) = \frac{1}{x + 3}$

$g \left(x\right) = - \frac{2}{x}$

The composition is

$\left[f o g\right] \left(x\right) = f \left(g \left(x\right)\right) = f \left(- \frac{2}{x}\right) = \frac{1}{\left(- \frac{2}{x}\right) + 3} = \frac{x}{3 x - 2}$

So,

$3 x - 2 \ne 0$, $x \ne \frac{2}{3}$

Therefore the domain of $f \left(g \left(x\right)\right)$ is ${D}_{f} \left(g \left(x\right)\right) = \mathbb{R} - \left\{\frac{2}{3}\right\}$