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?

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

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1 What are rational functions?

2 What can the graphs look like?

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