Graphing Radical Functions

Let's graph the function by making a table of values.

$x$	$\sqrt{x+8}-2$	$f(x)=\sqrt{x+8}-2$
${\color{#0000FF}{\text{-} 8}}$	$\sqrt{{\color{#0000FF}{\text{-} 8}}+8}-2$	${\color{#009600}{\text{-} 2}}$
${\color{#0000FF}{\text{-} 7}}$	$\sqrt{{\color{#0000FF}{\text{-} 7}}+8}-2$	${\color{#009600}{\text{-} 1}}$
${\color{#0000FF}{\text{-} 6}}$	$\sqrt{{\color{#0000FF}{\text{-} 6}}+8}-2$	$\approx {\color{#009600}{\text{-} 0.59}}$
${\color{#0000FF}{\text{-} 5}}$	$\sqrt{{\color{#0000FF}{\text{-} 5}}+8}-2$	$\approx {\color{#009600}{\text{-} 0.27}}$
${\color{#0000FF}{\text{-} 4}}$	$\sqrt{{\color{#0000FF}{\text{-} 4}}+8}-2$	${\color{#009600}{0}}$
${\color{#0000FF}{\text{-} 3}}$	$\sqrt{{\color{#0000FF}{\text{-} 3}}+8}-2$	$\approx {\color{#009600}{0.24}}$

Let's now plot the points and connect them with a smooth curve.

Domain

The domain of radical functions only includes values for which the ${\color{#FF0000}{\text{radicand}}}$ is non-negative. $\begin{gathered} f(x)=\sqrt{{\color{#FF0000}{x+8}}}-2 \end{gathered}$ We can solve for these values of $x$ by only looking at the radicand. $\begin{gathered} x+8 \geq 0 \quad \Leftrightarrow \quad x \geq \text{-} 8 \end{gathered}$ The domain is all real numbers greater than or equal to $\text{-} 8.$

Range

To identify the range, we will draw the graph of the function. We can see that the range is all real numbers greater than or equal to $\text{-} 2.$

Conclusion

Finally, let's summarize our findings. $\begin{aligned} \textbf{Domain:}&\ \ x \geq \text{-} 8 \\ \textbf{Range:}& \ \ f(x) \geq \text{-} 2 \end{aligned}$