5.5 Key features of rational functions (continued)

Summary

For each of the following rational functions, determine, with justification, the exact locations of all (i) horizontal asymptotes, (ii) vertical asymptotes, (iii) zeros, and (iv) holes of the function. Clearly show your work and thinking.

-$\displaystyle r(x) = \frac{-19(x+11.3)^2(x-15.1)(x-17.3)}{41(x+5.7)(x+11.3)(x-8.4)(x-15.1)}$-$\displaystyle s(x) = \frac{-29(x^2-16)(x^2+99)(x-53)}{101(x^2-4)(x-13)^2(x+104)}$-$\displaystyle u(x) = \frac{-71(x^2 - 13x + 36)(x-58.4)(x+78.2)}{83(x+58.4)(x-78.2)(x^2 - 12x + 27)}$

Find a formula for a rational function that meets the stated criteria, with justification. If no such formula is possible, explain why.

• A rational function $r(x)$ in the form $r(x) = \frac{k}{x-a} + b$ so that $r$ has a horizontal asymptote of $y = -\frac{3}{7}$, a vertical asymptote of $x = \frac{5}{2}$, and $r(0) = 4$. - A rational function $s(x)$ that has no horizontal asymptote, has zeros at $x = -5$ and $x = 3$, has a single vertical asymptote at $x = -1$, and satisfies $\lim_{x \to \infty} s(x) = -\infty$ and $\lim_{x \to -\infty} s(x) = +\infty$. - A rational function $u(x)$ that is positive for $x \lt -4$, negative for $-4 \lt x \lt -2$, negative for $-2 \lt x \lt 1$, positive for $1 \lt x \lt 5$, and negative for $x \gt 5$. The only zeros of $u$ are located at $x = -4$ and $x = -2$. In addition, $u$ has a hole at $x = 4$. - A rational function $w(x)$ whose graph is shown in Figure. A plot of the rational function $w$.

Graph each of the following rational functions and decide whether or not each function has an inverse function. If an inverse function exists, find its formula. In addition, state the domain and range of each function you consider (the original function as well as its inverse function, if the inverse function exists).

-$\displaystyle r(x) = -\frac{3}{x-4} + 5$-$\displaystyle s(x) = \frac{4 - 3x}{7x - 2}$-$\displaystyle u(x) = \frac{2x - 1}{(x-1)^2}$-$\displaystyle w(x) = \frac{11}{(x+4)^3} - 7$

For each of the following rational functions, identify the location of any potential hole in the graph. Then, create a table of function values for input values near where the hole should be located. Use your work to decide whether or not the graph indeed has a hole, with written justification.

-$\displaystyle r(x) = \frac{x^2-16}{x+4}$-$\displaystyle s(x) = \frac{(x-2)^2(x+3)}{x^2 - 5x - 6}$-$\displaystyle u(x) = \frac{(x-2)^3(x+3)}{(x^2 - 5x - 6)(x-7)}$-$\displaystyle w(x) = \frac{x^2 + x - 6}{(x^2 + 5x + 6)(x+3)}$- True or false: given $r(x) = \frac{p(x)}{q(x)}$, if $p(a) = 0$ and $q(a) = 0$, then $r$ has a hole at $x = a$.

In the questions that follow, we explore the average rate of change of power functions on the interval $[1,x]$. To begin, let $f(x) = x^2$ and let $A(x)$ be the average rate of change of $f$ on $[1,x]$.

• Explain why $A$ is a rational function of $x$. - What is the domain of $A$? - At the point where $A$ is undefined, does $A$ have a vertical asymptote or a hole? Justify your thinking clearly. - What can you say about the average rate of change of $f$ on $[1,x]$ as $x$ gets closer and closer (but not equal) to $1$? - Now let $g(x) = x^3$ and $B(x)$ be the average rate of change of $B$ on $[1,x]$. Respond to prompts (a) - (d) but this time for the function $B$ instead of $A$. - Finally, let $h(x) = x^4$ and $C(x)$ be the average rate of change of $C$ on $[1,x]$. Respond to prompts (a) - (d) but this time for the function $C$ instead of $A$.