5.6 Rational Functions

Determine the value for which the rational expression $\frac{x + 3}{2 x^{2} + 9 x - 5}$ is undefined

Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function.

After running out of pre-packaged supplies, a nurse in a refugee camp is preparing an intravenous sugar solution for patients in the camp hospital. A large mixing tank currently contains 100 gallons of distilled water into which 5 pounds of sugar have been mixed. A tap will open pouring 10 gallons per minute of distilled water into the tank at the same time sugar is poured into the tank at a rate…

There are 1,200 first-year and 1,500 second-year students at a rally at noon. After 12 p.m., 20 first-year students arrive at the rally every five minutes while 15 second-year students leave the rally. Find the ratio of first-year to second-year students at 1 p.m.

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

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

Find the vertical asymptotes and removable discontinuities of the graph of $f \left(x\right) = \frac{x^{2} - 25}{x^{3} - 6 x^{2} + 5 x} .$

In the sugar concentration problem earlier, we created the equation $C \left(t\right) = \frac{5 + t}{100 + 10 t} .$

Find the horizontal asymptote and interpret it in context of the problem.

Find the vertical and horizontal asymptotes of the function:

$f \left(x\right) = \frac{\left(2 x - 1\right) \left(2 x + 1\right)}{\left(x - 2\right) \left(x + 3\right)}$

Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a rational function. Then, find the x- and y-intercepts and the horizontal and vertical asymptotes.

What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?

If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?

Can a graph of a rational function have no x-intercepts? If so, how?

Find the domain of the rational functions.

$f \left(x\right) = \frac{x + 1}{x^{2} - 1}$

Find the domain of the rational functions.

$f \left(x\right) = \frac{x^{2} + 4 x - 3}{x^{4} - 5 x^{2} + 4}$

Find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

$f \left(x\right) = \frac{2}{5 x + 2}$

Find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

$f \left(x\right) = \frac{x}{x^{2} + 5 x - 36}$

Find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

$f \left(x\right) = \frac{3 x - 4}{x^{3} - 16 x}$

Find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

$f \left(x\right) = \frac{x + 5}{x^{2} - 25}$

Find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

$f \left(x\right) = \frac{4 - 2 x}{3 x - 1}$

Find the x- and y-intercepts for the functions.

$f \left(x\right) = \frac{x}{x^{2} - x}$

Find the x- and y-intercepts for the functions.

$f \left(x\right) = \frac{x^{2} + x + 6}{x^{2} - 10 x + 24}$

Describe the local and end behavior of the functions.

$f \left(x\right) = \frac{x}{2 x + 1}$

Describe the local and end behavior of the functions.

$f \left(x\right) = \frac{- 2 x}{x - 6}$

Describe the local and end behavior of the functions.

$f \left(x\right) = \frac{2 x^{2} - 32}{6 x^{2} + 13 x - 5}$

Find the slant asymptote of the functions.

$f \left(x\right) = \frac{4 x^{2} - 10}{2 x - 4}$

Find the slant asymptote of the functions.

$f \left(x\right) = \frac{6 x^{3} - 5 x}{3 x^{2} + 4}$

Write an equation for a rational function with the given characteristics.

Vertical asymptotes at $x = 5$ and $x = −5 ,$ x-intercepts at $\left(2 , 0\right)$ and $\left(−1 , 0\right) ,$ y-intercept at $\left(0 , 4\right)$

Write an equation for a rational function with the given characteristics.

Vertical asymptotes at $x = −4$ and $x = −5 ,$ x-intercepts at $\left(4 , 0\right)$ and $\left(−6 , 0\right) ,$ Horizontal asymptote at $y = 7$

Write an equation for a rational function with the given characteristics.

Vertical asymptote at $x = −1 ,$ Double zero at $x = 2 ,$ y-intercept at $\left(0 , 2\right)$

Make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

$f \left(x\right) = \frac{1}{x - 2}$

Make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

$f \left(x\right) = \frac{2 x}{x + 4}$

Make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

$f \left(x\right) = \frac{x^{2}}{x^{2} + 2 x + 1}$

Identify the removable discontinuity.

$f \left(x\right) = \frac{x^{2} - 4}{x - 2}$

Identify the removable discontinuity.

$f \left(x\right) = \frac{x^{2} + x - 6}{x - 2}$

Identify the removable discontinuity.

$f \left(x\right) = \frac{x^{3} + x^{2}}{x + 1}$

Express a rational function that describes the situation.

In the refugee camp hospital, a large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in…

Use the given rational function to answer the question.

The concentration $C$ of a drug in a patient’s bloodstream _$t$ hours after injection is given by $C \left(t\right) = \frac{100 t}{2 t^{2} + 75} .$ Use a calculator to approximate the time when the concentration is highest.

Construct a rational function that will help solve the problem. Then, use a calculator to answer the question.

A rectangular box with a square base is to have a volume of 20 cubic feet. The material for the base costs 30 cents/ square foot. The material for the sides costs 10 cents/square foot. The material for the top costs 20 cents/square foot. Determine the dimensions that will yield minimum cos…

Construct a rational function that will help solve the problem. Then, use a calculator to answer the question.

A right circular cylinder with no top has a volume of 50 cubic meters. Find the radius that will yield minimum surface area. Let $x$ = radius.

For the functions below, identify the horizontal or slant asymptote.

1. ⓐ $g \left(x\right) = \frac{6 x^{3} - 10 x}{2 x^{3} + 5 x^{2}}$
2. ⓑ $h \left(x\right) = \frac{x^{2} - 4 x + 1}{x + 2}$
3. ⓒ $k \left(x\right) = \frac{x^{2} + 4 x}{x^{3} - 8}$