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)=x356123x2243x2+10123xf(x) = \frac{x^{356} - 123x^{22} - 43x^{2}+10123}{x}

f(x)=(x1)(x2)(x3)(x4)(x3)(x2)(x1)(x2)(x4)(x2) 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 x satisfy the equation f(x)=1 f(x) = 1 ?

f(x)=x26x+62x4f(x)=\dfrac{x^2-6x+6}{2x-4} g(x)=ax2+bx+cxdg(x)=\dfrac{ax^2+bx+c}{x-d}

You are given two functions ff and gg above, where a,b,c,a, b, c, and dd are unknown constants. Also, you are given the following information about the function gg:

  • It has the same vertical asymptote as ff.

  • Its diagonal asymptote is perpendicular to that of ff, and these two asymptotes intersect each other on the yy-axis.

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

What is the value of the other xx-coordinate where ff and gg intersect?

Let xx be a real number. Consider the function

f(x)=x31x2+x2 f(x) = \frac{x^3 - 1}{x^2 + x - 2}

How many real zeroes does it have?


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