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Compute the domain of the real-valued function f(x) = sqrt(3 + sqrt(5 + sqrt(x)).

 Oct 10, 2022
 #1
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\(f(x)=\sqrt{3 + \sqrt{5 + \sqrt{x}}}\)

\(x\ge0\)

Smallest value of the function would be \(\sqrt{3 + \sqrt{5}}\)

Maximum value is infinity.

\(a + b\sqrt{c} = \sqrt{3 + \sqrt{5}}\)

\(a + b\sqrt{5} = \sqrt{3 + \sqrt{5}}\)

(a + bsqrt(5))^2 = 3 + sqrt(5)

a^2 + 5b^2 + 2absqrt(5) = 3 + sqrt(5)

a^2 + 5b^2 = 3

sqrt(5) = 2absqrt(5)

2ab = 1

ab = 1/2

 

You can do it from here... Domain: [a + bsqrt(c), inf)

Get a and b, or just use the sqrt(3 + sqrt(5))

 Oct 10, 2022

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