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Difference between revisions of "1978 AHSME Problems/Problem 17"

(Created page with "We are given that <cmath>[f(x^2 + 1)]^{\sqrt(x)} = k</cmath> We can rewrite <math>\frac{9+y^2}{y^2}</math> as <math>\frac{9}{y^2} + 1</math> Thus, our function is now <cmath>[...")
 
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== Problem 17 ==
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If <math>k</math> is a positive number and <math>f</math> is a function such that, for every positive number <math>x</math>, <math>\left[f(x^2+1)\right]^{\sqrt{x}}=k</math>;
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then, for every positive number <math>y</math>, <math>\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}</math> is equal to
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<math>\textbf{(A) }\sqrt{k}\qquad
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\textbf{(B) }2k\qquad
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\textbf{(C) }k\sqrt{k}\qquad
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\textbf{(D) }k^2\qquad
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\textbf{(E) }y\sqrt{k}  </math> 
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We are given that <cmath>[f(x^2 + 1)]^{\sqrt(x)} = k</cmath>
 
We are given that <cmath>[f(x^2 + 1)]^{\sqrt(x)} = k</cmath>
 
We can rewrite <math>\frac{9+y^2}{y^2}</math> as <math>\frac{9}{y^2} + 1</math>
 
We can rewrite <math>\frac{9+y^2}{y^2}</math> as <math>\frac{9}{y^2} + 1</math>
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~JustinLee2017
 
~JustinLee2017
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==See Also==
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{{AHSME box|year=1978|num-b=16|num-a=18}}
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{{MAA Notice}}

Revision as of 20:44, 13 February 2021

Problem 17

If $k$ is a positive number and $f$ is a function such that, for every positive number $x$, $\left[f(x^2+1)\right]^{\sqrt{x}}=k$; then, for every positive number $y$, $\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}$ is equal to

$\textbf{(A) }\sqrt{k}\qquad \textbf{(B) }2k\qquad \textbf{(C) }k\sqrt{k}\qquad \textbf{(D) }k^2\qquad \textbf{(E) }y\sqrt{k}$

We are given that \[[f(x^2 + 1)]^{\sqrt(x)} = k\] We can rewrite $\frac{9+y^2}{y^2}$ as $\frac{9}{y^2} + 1$ Thus, our function is now \[[f(\frac{9}{y^2} + 1)]^{\sqrt{\frac{12}{y}}} = k\] \[\Rrightarrow[f(\frac{9}{y^2} + 1)]^{\sqrt{\frac{3}{y} \cdot 4}} = k\] \[\Rrightarrow([f(\frac{9}{y^2} + 1)]^{\sqrt{\frac{3}{y}}})^{\sqrt{4}} = (k)^{\sqrt{4}}\] \[\Rrightarrow([f(\frac{9}{y^2} + 1)]^{\sqrt{\frac{3}{y}}})^{2} = (k)^{2} = k^2\]

\[\boxed{D}\]

~JustinLee2017


See Also

1978 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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