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## Graphing Radical Functions 1.1 - Solution

We want to graph the given cube root function. $\begin{gathered} f(x)=\dfrac{\sqrt[3]{x+6}}{2} \end{gathered}$ To do so, we will start by making a table of values. Recall that the radicand of a cube root can be any real number. Therefore, we can use any $x\text{-}$value to make the table. Let's start!

$x$ $\dfrac{\sqrt[3]{x+6}}{2}$ $f(x)=\dfrac{\sqrt[3]{x+6}}{2}$
${\color{#0000FF}{\text{-}15}}$ $\dfrac{\sqrt[3]{{\color{#0000FF}{\text{-}15}}+6}}{2}$ ${\color{#009600}{\text{-}1.040\ldots}}$
${\color{#0000FF}{\text{-}12}}$ $\dfrac{\sqrt[3]{{\color{#0000FF}{\text{-}12}}+6}}{2}$ ${\color{#009600}{\text{-}0.908\ldots}}$
${\color{#0000FF}{\text{-}9}}$ $\dfrac{\sqrt[3]{{\color{#0000FF}{\text{-}9}}+6}}{2}$ ${\color{#009600}{\text{-}0.721\dots}}$
${\color{#0000FF}{\text{-}6}}$ $\dfrac{\sqrt[3]{{\color{#0000FF}{\text{-}6}}+6}}{2}$ ${\color{#009600}{0}}$
${\color{#0000FF}{\text{-}3}}$ $\dfrac{\sqrt[3]{{\color{#0000FF}{\text{-}3}}+6}}{2}$ ${\color{#009600}{0.721\ldots}}$
${\color{#0000FF}{0}}$ $\dfrac{\sqrt[3]{{\color{#0000FF}{0}}+6}}{2}$ ${\color{#009600}{0.908\ldots}}$
${\color{#0000FF}{3}}$ $\dfrac{\sqrt[3]{{\color{#0000FF}{3}}+6}}{2}$ ${\color{#009600}{1.040\ldots}}$

The ordered pairs $({\color{#0000FF}{\text{-}15}},{\color{#009600}{\text{-}1.040}}),$ $({\color{#0000FF}{\text{-}12}},{\color{#009600}{\text{-}0.908}}),$ $({\color{#0000FF}{\text{-}9}},{\color{#009600}{0.721}}),$ $({\color{#0000FF}{\text{-}6}},{\color{#009600}{0}}),$ $({\color{#0000FF}{\text{-}3}},{\color{#009600}{0.721}}),$ $({\color{#0000FF}{0}},{\color{#009600}{0.908}}),$ and $({\color{#0000FF}{3}},{\color{#009600}{1.040}})$ all lie on the graph of the function. Now, we will plot and connect these points with a smooth curve.

Looking at the graph of the function, notice that it continues on to infinity in both the positive and negative directions. This means that the domain and range of the function are all real numbers. \begin{aligned} \textbf{Domain:}&\ \text{All real numbers.}\\ \textbf{Range:}&\ \text{All real numbers.} \end{aligned}