Difference between revisions of "2011 AMC 12A Problems/Problem 21"
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\textbf{(E)}\ 144 </math> | \textbf{(E)}\ 144 </math> | ||
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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>. | ||
Revision as of 12:51, 4 September 2024
Problem
Let 
, and for integers 
, let 
. If 
is the largest value of 
for which the domain of 
is nonempty, the domain of 
is ![$[c]$](http://latex.artofproblemsolving.com/5/2/b/52bae999d8320de9922bb4da965c7d6eb93f7ed1.png)
. What is 
?
Solution 1
The domain of is defined when 
.
![\[f_{2}(x)=f_{1}\left(\sqrt{4-x}\right)=\sqrt{1-\sqrt{4-x}}\]](http://latex.artofproblemsolving.com/4/5/3/4533f0f767c7fd6b4ecb2745c2749e1b019fd1f6.png)
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 must be nonnegative, we can see that the negative values of the previous domain will not work, so 
. Thus we now arrive at 
being the only number in the of domain of 
that defines 
. However, since we are looking for the largest value for for which the domain of is nonempty, we must continue checking until we arrive at a domain that is empty.
We continue with 
to get a domain of 
. 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. 
