Difference between revisions of "2003 AMC 12A Problems/Problem 25"
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(I got somewhere, someone else go the rest of the way.) |
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==Problem== | ==Problem== | ||
− | Let <math> | + | Let <math>f(x)= \sqrt{ax^2+bx} </math>. For how many [[real number | real]] values of <math>a</math> is there at least one [[positive number | positive]] value of <math> b </math> for which the [[domain]] of <math>f </math> and the [[range]] of <math> f </math> are the same [[set]]? |
<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ \mathrm{infinitely \ many} } </math> | <math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ \mathrm{infinitely \ many} } </math> | ||
== Solution== | == Solution== | ||
− | {{solution}} | + | The domain of this function is the range of the inverse function, and vice versa, so we find the inverse function: |
+ | |||
+ | <math>y=\sqrt{ax^2+bx}</math> | ||
+ | |||
+ | <math>y^2=ax^2+bx</math> | ||
+ | |||
+ | <math>x=\dfrac{-b\pm\sqrt{b^2+4ay^2}}{2a}</math> | ||
+ | |||
+ | The domain of this is all real <math>y</math> such that <math>4ay^2\geq -b^2</math> | ||
+ | |||
+ | The range of this function is the domain of the other function, which is all <math>x</math> such that <math>ax^2+bx\geq 0</math>. Thus we need to find all real <math>a</math> such that for all <math>x</math>, either both of those are true or neither are. | ||
+ | |||
+ | {{incomplete|solution}} | ||
==See Also== | ==See Also== |
Revision as of 09:09, 11 August 2008
Problem
Let . For how many real values of
is there at least one positive value of
for which the domain of
and the range of
are the same set?
Solution
The domain of this function is the range of the inverse function, and vice versa, so we find the inverse function:
The domain of this is all real such that
The range of this function is the domain of the other function, which is all such that
. Thus we need to find all real
such that for all
, either both of those are true or neither are.