QuestionFirst find $f+g, f-g, f g$, and $\frac{f}{g}$. Then determine the domain for each function. $$\begin{array}{l} f(x)=\sqrt{x+8} ; g(x)=\sqrt{x+2} \\ (f+g)(x)=\square \end{array}$$

Studdy Solution
Calculate $\left(\frac{f}{g}\right)(x)$:
$$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x+8}}{\sqrt{x+2}}$$
Determine the domain of $\left(\frac{f}{g}\right)(x)$:
In addition to $x \geq -2$, we need to ensure $g(x) \neq 0$, which means $\sqrt{x+2} \neq 0$. This implies $x+2 \neq 0$ or $x \neq -2$.
Thus, the domain is $x > -2$.
The expressions and domains are as follows: - $(f+g)(x) = \sqrt{x+8} + \sqrt{x+2}$ with domain $x \geq -2$. - $(f-g)(x) = \sqrt{x+8} - \sqrt{x+2}$ with domain $x \geq -2$. - $(fg)(x) = \sqrt{x+8} \cdot \sqrt{x+2}$ with domain $x \geq -2$. - $\left(\frac{f}{g}\right)(x) = \frac{\sqrt{x+8}}{\sqrt{x+2}}$ with domain $x > -2$.