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08 Jan 2025, 02:53
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If \(x\) is a positive number, \(f(x) = \sqrt{x}\), and \(f(f(x))\) is the reciprocal of a prime number, which of the following could be the value of \(x\)?
A. 0.25
B. 0.04
C. 0.016
D. 0.008
E. 0.0016
A. 0.25
B. 0.04
C. 0.016
D. 0.008
E. 0.0016
08 Jan 2025, 02:53
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Official Solution:
If \(x\) is a positive number, \(f(x) = \sqrt{x}\), and \(f(f(x))\) is the reciprocal of a prime number, which of the following could be the value of \(x\)?
A. 0.25
B. 0.04
C. 0.016
D. 0.008
E. 0.0016
Given \(f(x) = \sqrt{x}\), then:
\(f(\sqrt{x}) = \)
\(=\sqrt{\sqrt{x}} = \)
\(=\sqrt[4]{x} \)
Since \(\sqrt[4]{x} = \frac{1}{prime}\), taking this to the fourth power gives \(x = \frac{1}{prime^4}\).
Let's check the options:
A. \(0.25 = \frac{1}{4} =\frac{1}{2^2}\). Discard
B. \(0.04 =\frac{4}{100} = \frac{1}{25} = \frac{1}{5^2}\). Discard.
C. \(0.016 = \frac{16}{1,000} = \frac{2}{125} =\frac{2}{5^3}\). Discard.
D. \(0.008 = \frac{8}{1,000} =\frac{1}{125} = \frac{1}{5^3}\) Discard.
E. \(0.0016 = \frac{16}{10,000} =\frac{1}{625} =\frac{1}{5^4}\). Bingo!
Answer: E
If \(x\) is a positive number, \(f(x) = \sqrt{x}\), and \(f(f(x))\) is the reciprocal of a prime number, which of the following could be the value of \(x\)?
A. 0.25
B. 0.04
C. 0.016
D. 0.008
E. 0.0016
Given \(f(x) = \sqrt{x}\), then:
\(f(\sqrt{x}) = \)
\(=\sqrt{\sqrt{x}} = \)
\(=\sqrt[4]{x} \)
Since \(\sqrt[4]{x} = \frac{1}{prime}\), taking this to the fourth power gives \(x = \frac{1}{prime^4}\).
Let's check the options:
A. \(0.25 = \frac{1}{4} =\frac{1}{2^2}\). Discard
B. \(0.04 =\frac{4}{100} = \frac{1}{25} = \frac{1}{5^2}\). Discard.
C. \(0.016 = \frac{16}{1,000} = \frac{2}{125} =\frac{2}{5^3}\). Discard.
D. \(0.008 = \frac{8}{1,000} =\frac{1}{125} = \frac{1}{5^3}\) Discard.
E. \(0.0016 = \frac{16}{10,000} =\frac{1}{625} =\frac{1}{5^4}\). Bingo!
Answer: E
