# How do you find the domain of g(x) = root4(x-5)?

Jul 7, 2016

Set the argument equal to $0$ and solve. See below.

#### Explanation:

The domain of a function is the set of all $x$-values for which the function is defined. In other words, it's where the function exists.

In terms of radicals with even indexes (the index is that little number above the root, in this case $4$), the function is defined for all $x$ that make the argument (the stuff inside) positive or $0$. That's because you can't have a negative number inside a square root or fourth root or etcetera. For example, $\sqrt{- 1}$ is not defined. That implies that a number, when raised to the 4th power, equals $- 1$. Of course, that is impossible, since numbers raised to the 4th power are always positive.

All we have to do, then, is find out when $x - 5$ is greater than or equal to $0$. Expressed mathematically, we have:
$x - 5 \ge 0$

Solving, we see:
$x \ge 5$

So if $x$ is greater than or equal to $5$, we will have a non-negative fourth root and therefore the function will be defined for those values. The domain in interval notation is $\left[5 , \infty\right)$. You can confirm this by looking at the graph:
graph{root4(x-5) [-10, 10, -5, 5]}

Note how there's nothing for $x < 5$, because for those values, the radical is negative.