5.4 Rational Functions

Introduction

Long-range behavior of rational functions

The domain of a rational function

A drug company estimates that to produce a new drug, it will cost

2500$ to make.

Determine a formula for a function $C(q)$ that models the cost of producing $q$ grams of the drug. What familiar kind of function is $C$?

Note that we can multiply the formula for $r$ by the form of $1$ given by $1 = \frac{\frac{1}{x^2}}{\frac{1}{x^2}}$. Do so, and distribute and simplify as much as possible in both the numerator and denominator to write $r$ in a different algebraic form.

Having rewritten $r$, we are in a better position to evaluate $\lim_{x \to \infty} r(x)$. Using our work from (a), we have

. What is the exact value of this limit and why?

Next, determine

.

Use Desmos to plot $r$ on the interval $[-10,10]$. In addition, plot the horizontal line $y = \frac{3}{7}$. What is the meaning of the limits you found in (b) and (c)?

Using a similar algebraic approach to our work in Activity 5.4.1, multiply $s(x)$ by $1 = \frac{\frac{1}{x^2}}{\frac{1}{x^2}}$ and hence evaluate

. What value do you find?

Plot the function $y = s(x)$ on the interval $[-10,10]$. What is the graphical meaning of the limit you found in (a)?

Next, use appropriate algebraic work to consider $u(x)$ and evaluate

. What do you find?

Plot the function $y = u(x)$ on the interval $[-10,10]$. What is the graphical meaning of the limit you computed in (c)?

Determine the domain of the function $r(x) = \frac{5x^3 + 17x^2 - 9x + 4}{2x^3 - 6x^2 - 8x}$.

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

$\displaystyle g(x) = \frac{x^2 - 1}{x^2 + 3x - 4}$

$\displaystyle h(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2}$

$\displaystyle j(x) = \frac{(x+5)(x-3)(x+1)(x-4)}{(x+1)(x+3)(x-5)}$

$\displaystyle k(x) = \frac{2x^2 + 7}{3x^3 - 12x}$

$\displaystyle m(x) = \frac{5x^2 - 45}{7(x-2)(x-3)^2(x^2 + 9)(x+1)}$