 # Determine the domain of the function and prove that it is continuous on its doma vazelinahS 2021-10-20 Answered
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 $\left(-\mathrm{\infty },\mathrm{\infty }\right)$