Composition with a square root

Text

Let $f(x)=\sqrt{x}$ and $g(x)=x-4$.

• (a) Write $(f\circ g)(x)$ and find its domain.
• (b) Compute $(f\circ g)(13)$.

Solution

(a) $(f\circ g)(x)=f\big(g(x)\big)=\sqrt{x-4}$. It exists when the radicand is non-negative: $x-4\ge 0\Rightarrow x\ge 4,\qquad D=[4,\,+\infty).$ (b) $(f\circ g)(13)=\sqrt{13-4}=\sqrt{9}=3$. $\boxed{\,(f\circ g)(x)=\sqrt{x-4},\ D=[4,+\infty),\quad (f\circ g)(13)=3\,}$