Problem 10 · 2017 Math Kangaroo
Medium
Algebra & Patterns
substitution
The graph of which of the following functions has the most intersections with the graph of the function \(f(x) = x\)?
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Answer: B — \(g_2(x) = x^3\)
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Hint 1 of 2
Intersections with y = x happen where g(x) = x; count the real solutions for each option.
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Hint 2 of 2
Higher-degree powers can cross the line y = x more times, so check how many real roots each equation has.
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Approach: count real solutions of g(x) = x for each choice
- Set each function equal to x. x^2 = x gives 2 solutions; x^3 = x gives 3 (x = -1,0,1).
- x^4 = x gives 2; -x^4 = x gives 2; -x = x gives 1.
- The most intersections is 3, from g2(x) = x^3.
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