Key Concepts

• We can use arrow notation to describe local behavior and end behavior of the toolkit functions [latex]f\left(x\right)=\frac{1}{x}[/latex] and [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex].
• A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote.
• Application problems involving rates and concentrations often involve rational functions.
• The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.
• 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.
• A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero.
• A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.
• Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior.
• If a rational function has x-intercepts at [latex]x={x}_{1},{x}_{2},\dots ,{x}_{n}[/latex], vertical asymptotes at [latex]x={v}_{1},{v}_{2},\dots ,{v}_{m}[/latex], and no [latex]{x}_{i}=\text{any }{v}_{j}[/latex], then the function can be written in the form [latex]f\left(x\right)=a\frac{{\left(x-{x}_{1}\right)}^{{p}_{1}}{\left(x-{x}_{2}\right)}^{{p}_{2}}\cdots {\left(x-{x}_{n}\right)}^{{p}_{n}}}{{\left(x-{v}_{1}\right)}^{{q}_{1}}{\left(x-{v}_{2}\right)}^{{q}_{2}}\cdots {\left(x-{v}_{m}\right)}^{{q}_{n}}}[/latex]