Perpendicular Asymptotes

Algebra Level 3

\[f(x)=\dfrac{x^2-6x+6}{2x-4}\] \[g(x)=\dfrac{ax^2+bx+c}{x-d}\]

You are given two functions \(f\) and \(g\) above, where \(a, b, c,\) and \(d\) are unknown constants. Also, you are given the following information about the function \(g\):

  • It has the same vertical asymptote as \(f\).

  • Its diagonal asymptote is perpendicular to that of \(f\), and these two asymptotes intersect each other on the \(y\)-axis.

  • The graphs of \(f\) and \(g\) have two intersection points. One of them is at \(x = -2\). (In other words, \(f(-2) = g(-2)\).)

What is the value of the other \(x\)-coordinate where \(f\) and \(g\) intersect?


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