# What is the domain of f(g(x)) if f(x)=x^2-4 and g(x)=sqrt(2x+4) ?

Aug 9, 2015

$x > - 2$

#### Explanation:

The domain of every function $f \left(x\right)$ is the set of $x$-values that are 'plugged' into the function $f$. It then follows that the domain of $f \left(u\right)$ is the set of $u$-values plugged into the function $f$. Make the substitution $u = g \left(x\right)$. The domain of $g \left(x\right)$ determines the set of $u$-values that are plugged into $f \left(x\right)$.

In short
Domain of $g \left(x\right)$$\left(g\right) \to$ Range of $g \left(x\right)$ = Domain of $f \left(u\right)$$\left(f\right) \to$ Range of $f \left(u\right)$ = Range of $f \left(g \left(x\right)\right)$

Thus the domain of $f \left(g \left(x\right)\right)$ = set of $x$-values that are plugged into the $f g$ function = set of $x$-values that are plugged into the $g$ function = domain of $g \left(x\right)$ = $x > - 2$ (for real values of $\sqrt{2 x + 4}$, $2 x + 4 > 0 \setminus R i g h t a r r o w x > - 2$