5.11: Rational functions

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1. Bronze is an alloy, or mixture, of copper and tin. The alloy initially contains 3 kg of copper and 9 kg tin. You add $x \mathrm{~kg}$ of copper to this 12 kg of alloy. The concentration of copper in the alloy is a function of $x$ :

\[(\text { Concentration of copper })=f(x)=\frac{\text { Total amount of copper }}{\text { Total amount of alloy }} .\]

[(a)] Find a formula for $f$ in terms of $x$, the amount of copper added.

[(b)] Evaluate the following expressions and explain their significance for the alloy:

[i.] $f\left(\frac{1}{2}\right)$

[ii.] $f(0)$

[iii.] $f(-1)$

[iv.] $f^{-1}\left(\frac{1}{2}\right)$

[v.] $f^{-1}(0)$

[(c)] Graph $f(x)$ using Desmos. Interpret the intercepts in the context of the alloy.

[(d)] Zoom out so you can see a larger viewing window. Does the appearance of the graph agree with what you expect to happen when large amounts of copper are added to the alloy?

2. The total cost $C(n)$ for a producer to manufacture $n$ units of a good is given by

\[C(n)=5000+50 n .\]

The average cost of producing $n$ units is $a(n)=\frac{C(n)}{n}$.

[(a)] Evaluate $C(1), C(100), C(1000)$, and $C(10000)$. Explain in a sentence the economic significance of these values.

[(b)] Evaluate: $a(1), a(100), a(1000)$, and $a(10000)$. Explain in a sentence the economic significance of these values.

[(c)] Based on part (b), what trend do you notice in the values of $a(n)$ as $n$ gets large? Explain this trend in economic terms.

3. Knowing that we can express a rational function as $r(x)=\frac{p(x)}{q(x)}$, answer the questions below.

[(a)] If $f(x)=\frac{x^2+1}{-3 x^2+5}$, what is $p(x)$ and what is $q(x)$ ?

[(b)] What are the leading terms of $p$ and $q$ for this rational function?

[(c)] To determine the long-run behavior of a rational function, we can simply examine the function $y=\frac{\text { Leading term of } p}{\text { Leading term of } q}$. What is the new rational function we can use to examine the long-run behavior of $f(x)$ ?

[(d)] What does this new function tell us about the long-run behavior of $f(x)$ ?

[(e)] Graph $f(x)$ using Desmos. Does the graph match your answer to part (f)?

[4.] Using the same steps from problem 1, determine the long-run behaviors of the following:

[(a)] $f(x)=\frac{2 x^2+1}{x^2+5}$

[(b)] $g(x)=\frac{x^3+1}{x^2+5}$

[(c)] $h(x)=\frac{x+1}{x^2+5}$

[(d)] Check your work on each part above by using a calculator.

[(e)] Compare the long-run behavior of $f(x), g(x)$, and $h(x)$. What do you notice? Discuss.

[5.] Let $m(x)=\frac{-x}{x+1}+\frac{x}{x+2}$.

[(a)] Write $m(x)$ as a rational function.

[(b)] Determine the long-run behavior of $m(x)$.

[6.] Let $f(x)=\frac{3 x^2+x}{2 x^2+5 x^3}$

[(a)] Graph $f(x)$ using Desmos.

(b)] Determine the long-run behavior of $\mathrm{f}(\mathrm{x})$ ?

[7.] Let $g(x)=\frac{1}{1+\frac{1}{x}}$.

[(a)] Rewrite $g(x)$ as a standard rational function.

[(b)] Using what you have learned in previous questions, find the horizontal asymptote, if it exists, of $g(x)$.

\item[8.] It costs a company \$30,000 to begin production of a good, plus \$3 for every unit of the good produced. Let $x$ be the number of units produced by the company.

[(a)] Find a formula fo $C(x)$, the total cost for the production of $x$ units of the good.

[(b)] Find a formula for the company's average cost per unit, $a(x)$.

[(c)] Graph $y=a(x)$. What is the horizontal asymptote?

[(d)] Explain in economic terms why the graph of $a$ has the long-run behavior that it does.

[(e)] Explain in economic terms why the graph of $a$ has the vertical asymptote that it does.

[(f)] Find a formula for $a^{-1}(y)$. Give an economic interpretation of $a^{-1}(y)$.

[(g)] The company makes a profit if the average cost of its good is less than \$5 per unit. Find the minimum number of units the company can produce and make a profit.

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