Key to "Determine the domain of the function and prove..."

vazelinahS

Answered question

2021-10-20

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 (x) = {(x^{4} + 1)}^{\frac{3}{2}}

Answer & Explanation

nitruraviX

Skilled2021-10-21Added 101 answers

Given:
$f (x) = {(x^{4} + 1)}^{\frac{3}{2}}$
The function has no undefined points nor domain constraints. Therefore, the is $- \infty < x < \infty$
Interval notation $\to (- \infty, \infty)$
By law of continuity:
$lim_{x \to \infty} {(x^{4} + 1)}^{\frac{3}{2}}$
$lim_{x \to a} {[f (x)]}^{b} = [lim_{x \to a} f (x)}]^{b}$
${(lim_{x \to \infty} (x^{4} + 1))}^{\frac{3}{2}}$
Apply Infinity Property: $lim_{x \to \pm \infty} (a x^{n} + \dots + b x + c) = \infty, a > 0$ n is even
a=1, n=4
$= \infty^{\frac{3}{2}}$
Apply Infinity Property: $\infty^{c} = \infty$
$= \infty$
Hence proved
f(x) is continuous in its domain $(- \infty, \infty)$

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