2003 AMC 12A Problems/Problem 25

Problem

Let $f(x)= \sqrt{ax^2+bx}$ . For how many real values of $a$ is there at least one positive value of $b$ for which the domain of $f$ and the range of $f$ are the same set?

$\mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ \mathrm{infinitely \ many} }$

Solution

The domain of this function is the range of the inverse function, and vice versa, so we find the inverse function:

$y=\sqrt{ax^2+bx}$

$y^2=ax^2+bx$

$x=\dfrac{-b\pm\sqrt{b^2+4ay^2}}{2a}$

The domain of this is all real $y$ such that $4ay^2\geq -b^2$

The range of this function is the domain of the other function, which is all $x$ such that $ax^2+bx\geq 0$ . Thus we need to find all real $a$ such that for all $x$ , either both of those are true or neither are.

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2003 AMC 12A Problems/Problem 25

Problem

Solution

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