3.7 Rational functions  (Page 10/16)

Key equations

Key concepts

• We can use arrow notation to describe local behavior and end behavior of the toolkit functions $f(x)=\frac{1}{x}$ and $f(x)=\frac{1}{x^2}.$ See [link] .
• A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. See [link] .
• Application problems involving rates and concentrations often involve rational functions. See [link] .
• The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. See [link] .
• The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. See [link] .
• A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. See [link] .
• A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. See [link] , [link] , [link] , and [link] .
• Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. See [link] .
• If a rational function has x -intercepts at $x=x_1,x_2,\dots ,x_n,$ vertical asymptotes at $x=v_1,v_2,\dots ,v_m,$ and no $x_i=\text{any }{v}_j,$ then the function can be written in the form
  $\begin{array}{l}\begin{array}{l}\hfill \\ f(x)=a\frac{{(x-x_1)}^{p_1}{(x-x_2)}^{p_2}\cdots {(x-x_n)}^{p_n}}{{(x-v_1)}^{q_1}{(x-v_2)}^{q_2}\cdots {(x-v_m)}^{q_n}}\hfill \end{array}\hfill \end{array}$

  See [link] .

Section exercises

Verbal

Algebraic

For the following exercises, find the domain of the rational functions.

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

For the following exercises, find the x - and y -intercepts for the functions.

For the following exercises, describe the local and end behavior of the functions.