Rational Functions

Rational Functions: Level 2 Challenges


True or false:

All rational functions have some form of asymptote, whether it's slant, vertical, or horizontal.

How many vertical asymptotes are there on the following rational function?

\[f(x) = \frac{x^{356} - 123x^{22} - 43x^{2}+10123}{x}\]

\[ f(x) = \dfrac{(x-1)(x-2)(x-3)(x-4)(x-3)(x-2)(x-1)}{(x-2)(x-4)(x-2)}\]

How many values of \( x \) satisfy the equation \( f(x) = 1 \) ?

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

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?

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?


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