Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Find the answer

5. The function defined on the domain $R$
$f(x)=|\lg | x-2||-1 \text{. }$

If $b<0$ , then the equation concerning $x$
$f^{2}(x)+b f(x)=0$

has $\qquad$ distinct real roots.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

5. 8 .

From the problem, we know that the graph of $y=\lg x$ is first symmetrical about the $y$ -axis, then the part below the $x$ -axis is flipped up, followed by a rightward shift of 2 units and a downward shift of 1 unit to form the graph of $f(x)$ . The zeros of $g(x)=f^{2}(x)+b f(x)$ are the roots of the equation. It is easy to see that $g(x)$ has 8 zeros.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.