What is the domain of x^(1/3)?

Jul 30, 2018

$x \in \mathbb{R}$

Explanation:

The domain is the set of $x$ values that make this function defined. We have the following:

$f \left(x\right) = {x}^{\frac{1}{3}}$

Is there any $x$ that will make this function undefined? Is there anything that we cannot raise to the one-third power?

No! We can plug in any value for $x$ and get a corresponding $f \left(x\right)$.

To make this more tangible, let's plug in some values for $x$:

$x = 27 \implies f \left(27\right) = {27}^{\frac{1}{3}} = 3$

$x = 64 \implies f \left(64\right) = {64}^{\frac{1}{3}} = 4$

$x = 2187 \implies f \left(2187\right) = {2187}^{\frac{1}{3}} = 7$

$x = 5000 \implies f \left(5000\right) = {5000}^{\frac{1}{3}} \approx 17.1$

Notice, I could have used much higher $x$ values, but we got an answer each time. Thus, we can say our domain is

$x \in \mathbb{R}$, which is just a mathy way of saying $x$ can take on any value.

Hope this helps!