Difference between revisions of "2011 AMC 12A Problems/Problem 21"
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The domain of <math>f_{1}(x)=\sqrt{1-x}</math> is defined when <math>x\leq1</math>. | The domain of <math>f_{1}(x)=\sqrt{1-x}</math> is defined when <math>x\leq1</math>. | ||
− | <cmath>f_{2}(x)=f_{1}(\sqrt{4-x})=\sqrt{1-\sqrt{4-x}}</cmath> | + | <cmath>f_{2}(x)=f_{1}\left(\sqrt{4-x}\right)=\sqrt{1-\sqrt{4-x}}</cmath> |
Applying the domain of <math>f_{1}(x)</math> and the fact that square roots must be positive, we get <math>0\leq\sqrt{4-x}\leq1</math>. Simplifying, the domain of <math>f_{2}(x)</math> becomes <math>3\leq x\leq4</math>. | Applying the domain of <math>f_{1}(x)</math> and the fact that square roots must be positive, we get <math>0\leq\sqrt{4-x}\leq1</math>. Simplifying, the domain of <math>f_{2}(x)</math> becomes <math>3\leq x\leq4</math>. |
Revision as of 21:15, 4 August 2021
Problem
Let , and for integers , let . If is the largest value of for which the domain of is nonempty, the domain of is . What is ?
Solution
The domain of is defined when .
Applying the domain of and the fact that square roots must be positive, we get . Simplifying, the domain of becomes .
Repeat this process for to get a domain of .
For , since square roots are positive, we can exclude the negative values of the previous domain to arrive at as the domain of . We now arrive at a domain with a single number that defines , however, since we are looking for the largest value for for which the domain of is nonempty, we must continue until we arrive at a domain that is empty. We continue with to get a domain of . Solve for to get . Since square roots cannot be negative, this is the last nonempty domain. We add to get .
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.