Rational functions

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Rational functions are functions defined by fractions of polynomials.

1 What are rational functions?

A rational function in the variable $x$ is a function the form

\[ f(x) = \frac {p(x)}{q(x)} \]

where $p$ and $q$ are polynomial functions. The domain of a rational function is all real numbers except for where the denominator is equal to zero.

Which of the following are rational functions?

$f(x) = 0$ $f(x) = \frac {3x+1}{x^2-4x+5}$ $f(x)=e^x$ $f(x)=\frac {\sin (x)}{\cos (x)}$ $f(x) = -4x^{-3}+5x^{-1}+7-18x^2$ $f(x) = x^{1/2}-x +8$ $f(x)=\frac {\sqrt {x}}{x^3-x}$

All polynomials can be thought of as rational functions.

2 What can the graphs look like?

There is a somewhat wide variation in the graphs of rational functions.

Here we see the the graphs of four rational functions.

Match the curves $A$, $B$, $C$, and $D$ with the functions \begin{align*} &\frac {x^2-3x+2}{x-2}, &&\frac {x^2-3x+2}{x+2}, \\ &\frac {x-2}{x^2-3x+2}, &&\frac {x+2}{x^2-3x+2}. \end{align*}

Consider $\frac {x^2-3x+2}{x-2}$. This function is undefined only at $x=2$. Of the curves that we see above, $\answer [given]{D}$ is undefined exactly at $x=2$.

Now consider $\frac {x^2-3x+2}{x+2}$. This function is undefined only at $x=-2$. The only function above that undefined exactly at $x=-2$ is curve $\answer [given]{A}$.

Now consider $\frac {x-2}{x^2-3x+2}$. This function is undefined at the roots of

\[ x^2-3x+2 = (x-2)(x-1). \]

Hence it is undefined at $x=2$ and $x=1$. It looks like both curves $B$ and $C$ would work. Distinguishing between these two curves is easy enough if we evaluate at $x=-2$. Check it out. \begin{align*} \bigg [\frac {x-2}{x^2-3x+2}\bigg ]_{x=-2} &= \frac {-2-2}{(-2)^2-3(-2)+2}\\ &= \frac {-4}{4+6+2}\\ &=\frac {-4}{12}. \end{align*}

Since this is negative, we see that $\frac {x-2}{x^2-3x+2}$ corresponds to curve $\answer [given]{B}$.

Finally, it must be the case that curve $\answer [given]{C}$ corresponds to $\frac {x+2}{x^2-3x+2}$. We should note that if this function is evaluated at $x=-2$, the output is zero, and this corroborates our work above.

2025-01-06 19:43:55