# 5.6 Rational functions  (Page 3/16)

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## Rational function

A rational function    is a function that can be written as the quotient of two polynomial functions

$f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{a}_{p}{x}^{p}+{a}_{p-1}{x}^{p-1}+...+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q-1}{x}^{q-1}+...+{b}_{1}x+{b}_{0}},Q\left(x\right)\ne 0$

## Solving an applied problem involving a rational function

A large mixing tank currently contains 100 gallons of water into which 5 pounds of sugar have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of 1 pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after 12 minutes. Is that a greater concentration than at the beginning?

Let $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ be the number of minutes since the tap opened. Since the water increases at 10 gallons per minute, and the sugar increases at 1 pound per minute, these are constant rates of change. This tells us the amount of water in the tank is changing linearly, as is the amount of sugar in the tank. We can write an equation independently for each:

The concentration, $\text{\hspace{0.17em}}C,\text{\hspace{0.17em}}$ will be the ratio of pounds of sugar to gallons of water

$C\left(t\right)=\frac{5+t}{100+10t}$

The concentration after 12 minutes is given by evaluating $\text{\hspace{0.17em}}C\left(t\right)\text{\hspace{0.17em}}$ at

$\begin{array}{ccc}\hfill C\left(12\right)& =& \frac{5+12}{100+10\left(12\right)}\hfill \\ & =& \frac{17}{220}\hfill \end{array}$

This means the concentration is 17 pounds of sugar to 220 gallons of water.

At the beginning, the concentration is

$\begin{array}{ccc}\hfill C\left(0\right)& =& \frac{5+0}{100+10\left(0\right)}\hfill \\ & =& \frac{1}{20}\hfill \end{array}$

Since $\text{\hspace{0.17em}}\frac{17}{220}\approx 0.08>\frac{1}{20}=0.05,\text{\hspace{0.17em}}$ the concentration is greater after 12 minutes than at the beginning.

There are 1,200 freshmen and 1,500 sophomores at a prep rally at noon. After 12 p.m., 20 freshmen arrive at the rally every five minutes while 15 sophomores leave the rally. Find the ratio of freshmen to sophomores at 1 p.m.

$\frac{12}{11}$

## Finding the domains of rational functions

A vertical asymptote    represents a value at which a rational function is undefined, so that value is not in the domain of the function. A reciprocal function cannot have values in its domain that cause the denominator to equal zero. In general, to find the domain of a rational function, we need to determine which inputs would cause division by zero.

## Domain of a rational function

The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.

Given a rational function, find the domain.

1. Set the denominator equal to zero.
2. Solve to find the x -values that cause the denominator to equal zero.
3. The domain is all real numbers except those found in Step 2.

## Finding the domain of a rational function

Find the domain of $\text{\hspace{0.17em}}f\left(x\right)=\frac{x+3}{{x}^{2}-9}.$

Begin by setting the denominator equal to zero and solving.

$\begin{array}{ccc}\hfill {x}^{2}-9& =& 0\hfill \\ \hfill {x}^{2}& =& 9\hfill \\ \hfill x& =& ±3\hfill \end{array}$

The denominator is equal to zero when $\text{\hspace{0.17em}}x=±3.\text{\hspace{0.17em}}$ The domain of the function is all real numbers except $\text{\hspace{0.17em}}x=±3.$

Find the domain of $\text{\hspace{0.17em}}f\left(x\right)=\frac{4x}{5\left(x-1\right)\left(x-5\right)}.$

The domain is all real numbers except $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=5.$

## Identifying vertical asymptotes of rational functions

By looking at the graph of a rational function, we can investigate its local behavior and easily see whether there are asymptotes. We may even be able to approximate their location. Even without the graph, however, we can still determine whether a given rational function has any asymptotes, and calculate their location.

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