# 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.