# Determine the domain of the function and prove that it is continuous on its doma

Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section. $f\left(x\right)={\left({x}^{4}+1\right)}^{\frac{3}{2}}$
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Given:
$f\left(x\right)={\left({x}^{4}+1\right)}^{\frac{3}{2}}$
The function has no undefined points nor domain constraints. Therefore, the is $-\mathrm{\infty }
Interval notation $\to \left(-\mathrm{\infty },\mathrm{\infty }\right)$
By law of continuity:
$\underset{x\to \mathrm{\infty }}{lim}{\left({x}^{4}+1\right)}^{\frac{3}{2}}$
$\underset{x\to a}{lim}{\left[f\left(x\right)\right]}^{b}=\left[\underset{x\to a}{lim}f\left(x\right)\right\}{\right]}^{b}$
${\left(\underset{x\to \mathrm{\infty }}{lim}\left({x}^{4}+1\right)\right)}^{\frac{3}{2}}$
Apply Infinity Property: $\underset{x\to ±\mathrm{\infty }}{lim}\left(a{x}^{n}+\dots +bx+c\right)=\mathrm{\infty },a>0$ n is even
a=1, n=4
$={\mathrm{\infty }}^{\frac{3}{2}}$
Apply Infinity Property: ${\mathrm{\infty }}^{c}=\mathrm{\infty }$
$=\mathrm{\infty }$
Hence proved
f(x) is continuous in its domain $\left(-\mathrm{\infty },\mathrm{\infty }\right)$