WMC 2018 Pursuit - Pack 1, A1

Algebra Level 2

For x > 0 x>0 , let f ( x ) = x 3 − 3 x 2 − 9 x f(x)=x^3-3x^2-9x . Find the sum of all values of x x for all the points of intersection of the graphs of f ( x ) f(x) and its inverse operation.

WMC Pursuit Questions


The answer is 5.

Julian Yu by Julian Yu , 19, Philippines

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1 solution

When the f ( x ) f(x) intersects with its inverse f − 1 ( x ) f^{-1}(x) , it implies that

f ( x ) = x x 3 − 3 x 2 − 9 x = x x ( x 2 − 3 x − 10 ) = 0 x ( x + 2 ) ( x − 5 ) = 0 ⟹ x = { − 2 0 5 \begin{aligned} f(x) & = x \\ x^3-3x^2-9x & = x \\ x(x^2-3x-10) & = 0 \\ x(x+2)(x-5) & = 0 \\ \implies x & = \begin{cases} -2 \\ 0 \\ 5 \end{cases} \end{aligned}

For x > 0 x >0 , the answer is 5 \boxed 5 .

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