Graphing Radical Functions

We want to graph the given radical function. $$\begin{array}{c}f(x)=\sqrt{x-4}-10\end{array}$$ To do so, we will start by making a table of values. Recall that the radicand cannot be negative. With this in mind, we can first determine its domain as shown below. $$\begin{array}{c}x-4\ge 0\iff x\ge 4\end{array}$$ Therefore, we will use $x\text{-}$values greater than or equal to $4.$ Let's start!

The ordered pairs $(4,\text{-}10),$ $(5,\text{-}9),$ $(6,\text{-}8.585),$ $(7,\text{-}8.267),$ $(8,\text{-}8),$ and $(9,\text{-}7.763)$ all lie on the graph of the function. Now, we will plot and connect these points with a smooth curve.

We can see above that the graph of the function goes to infinity in the positive direction starting from $y=\text{-}10.$ Therefore, its range is all real numbers greater than or equal to $\text{-}10.$ $$\begin{array}{rl}\text{Domain:}& x\ge 4\\ \text{Range:}& y\ge \text{-}10\end{array}$$