Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 25

Prove that $h(x) = \sqrt[4]{x} + x^3 \cos x$ is continuous on its domain. State the domain.

We can rewrite $h(x) = \sqrt[4]{x} + x^3 \cos x$ as

$\qquad h(x) = F(x) + G(x) \cdot I(x)$

where

$\qquad F(x) = \sqrt[4]{x}, \qquad G(x) = x^3 \qquad$ and $\qquad I(x) = \cos x$

$\qquad G(x)$ and $I(x)$ are one example of function that is continuous on every values of $x$ according to the definition. However, $F(x)$ is a root function that is only restricted on its domain $[0, \infty)$

Therefore,

$\qquad$ The domain of $h(x)$ is $[0, \infty)$