Rational Functions

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Introduction

James: Hey guys, I slept through class yesterday... could you fill me in on what a rational function is?
Julia: See, class didn’t make a lot of sense to me because I was thinking, ‘‘Functions can be rational?’’
Dylan: They don’t mean rational like me or you, Julia! It means the function can be represented as a fraction where the numerator and denominator are both polynomials.
Julia and James: Oh!
Dylan: Rational functions are pretty neat, because they can have two different types of discontinuities!
Altogether: LET’S DIVE IN!

Guided Example

Consider the function $f(x)=\dfrac {(x-2)(x+4)}{(x-3)(x+3)(x+4)}$

Describe the graph. What strange things do you notice?

The ‘‘hole’’ present in the graph is called a removable discontinuity.

The vertical lines that the curve approaches at $x=3$ and $x=-3$ is called a vertical asymptote, another type of discontinuity.

On Your Own

Find and report the locations of discontinuities in the following functions, note that at this time Desmos does not show removable discontinuities. You will need to find those by hand:

$a(x) = \dfrac {x^2+1}{x-2}$ $\graph {}$

$x=2$ $x=-1$ $x=1$ $x=0$ $x=4$ None

$b(x) = \dfrac {x^2-5x+7}{x^2-x-6}$ $\graph {}$

$x=2$ $x=-2$ $x=1$ $x=0$ $x=3$ None

$c(x) = \dfrac {x^2-x}{x}$

$\graph {}$

$x=2$ $x=5$ $x=1$ $x=0$ $x=3$ None

$d(x) = \dfrac {x+2}{x^2-x-6}$

$\graph {}$

$x=-6$ $x=3$ $x=2$ $x=0$ $x=-2$ None

$f(x) = \dfrac {2x^2+5}{x^2-25}$

$\graph {}$

$x=-1$ $x=-5$ $x=10$ $x=0$ $x=5$ None

$g(x) = \dfrac {x^3-x^2-15x-9}{x+3}$

$\graph {}$

$x=12$ $x=-4$ $x=3$ $x=5$ $x=-3$ None

$h(x) = \dfrac {1}{3x^2-x}$

$\graph {}$

$x=-\sqrt {3}$ $x=\frac {1}{3}$ $x=3$ $x=0$ $x=-3$ None

How can you tell if a rational function has a vertical asymptote or a removable discontinuity?

Vertical asymptotes occur where only the denominator approaches zero, and removable discontinuities

occur where both the numerator and denominator approach zero.

Vertical asymptotes occur where only the numerator approaches zero, and removable discontinuities

occur where both the numerator and denominator approach zero.

Vertical asymptotes occur where both the numerator and denominator approach zero, and removable

discontinuities occur where only the denominator approaches zero.

Vertical asymptotes occur where only the numerator approaches zero, and removable

discontinuities occur where only the denominator approaches zero.

In Summary

James: These functions are pretty neat! What were they called again?
Dylan: They’re called rational functions, fractions where the numerator and denominator are both polynomials!
Julia: So, when exactly does a vertical asymptote occur?
James: I know this one! Vertical asymptotes occur at points where the denominator of the function will be zero, but the numerator is non-zero!
Julia: That makes sense! But when do removable discontinuities occur then?
Dylan: Removable discontinuities occur where the numerator and denominator are both zero.