Answer
a. $(-\infty,\infty)$
b. $[0,\infty)$
c. $(-\infty,\infty)$
d. $(0,\infty)$
Work Step by Step
a. The function $f(x)=x^{1/3}$ has a domain of $(-\infty,\infty)$ and $\lim_{x\to c}f(x)=\lim_{x\to c}x^{1/3}=c^{1/3}=f(c)$ for every $c$ in the entire domain; thus the function is continuous over $(-\infty,\infty)$
b. The function $g(x)=x^{3/4}$ has a domain of $[0,\infty)$ and $\lim_{x\to c}g(x)=\lim_{x\to c}x^{3/4}=c^{3/4}=f(c)$ for every $c$ in the domain; thus the function is continuous over $[0,\infty)$
c. The function $h(x)=x^{-2/3}$ has a domain of $(-\infty,\infty)$ and $\lim_{x\to c}f(x)=\lim_{x\to c}x^{-2/3}=c^{-2/3}=f(c)$ for every $c$ in the entire domain; thus the function is continuous over $(-\infty,\infty)$
d. The function $k(x)=x^{-1/6}$ has a domain of $(0,\infty)$ and $\lim_{x\to c}g(x)=\lim_{x\to c}x^{-1/6}=c^{-1/6}=f(c)$ for every $c$ in the domain; thus the function is continuous over $(0,\infty)$
