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Rational Functions
True or false:
All rational functions have some form of asymptote, whether it's slant, vertical, or horizontal.
by
Hobart Pao
How many vertical asymptotes are there on the following rational function?
\[f(x) = \frac{x^{356} - 123x^{22} - 43x^{2}+10123}{x}\]
by
Spencer Whitehead
\[ f(x) = \dfrac{(x-1)(x-2)(x-3)(x-4)(x-3)(x-2)(x-1)}{(x-2)(x-4)(x-2)}\]
How many values of \( x \) satisfy the equation \( f(x) = 1 \) ?
by
Pranshu Gaba
\[f(x)=\dfrac{x^2-6x+6}{2x-4}\] \[g(x)=\dfrac{ax^2+bx+c}{x-d}\]
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?
by
Arjen Vreugdenhil
Let \(x\) be a real number. Consider the function
\[ f(x) = \frac{x^3 - 1}{x^2 + x - 2} \]
How many real zeroes does it have?
by
Denton Young