5.6 Rational functions  (Page 10/16)

 Page 10 / 16

Key equations

 Rational Function

Key concepts

• We can use arrow notation to describe local behavior and end behavior of the toolkit functions $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{x}^{2}}.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}x={x}_{1},{x}_{2},\dots ,{x}_{n},\text{\hspace{0.17em}}$ vertical asymptotes at $\text{\hspace{0.17em}}x={v}_{1},{v}_{2},\dots ,{v}_{m},\text{\hspace{0.17em}}$ and no then the function can be written in the form
$\begin{array}{l}\begin{array}{l}\hfill \\ f\left(x\right)=a\frac{{\left(x-{x}_{1}\right)}^{{p}_{1}}{\left(x-{x}_{2}\right)}^{{p}_{2}}\cdots {\left(x-{x}_{n}\right)}^{{p}_{n}}}{{\left(x-{v}_{1}\right)}^{{q}_{1}}{\left(x-{v}_{2}\right)}^{{q}_{2}}\cdots {\left(x-{v}_{m}\right)}^{{q}_{n}}}\hfill \end{array}\hfill \end{array}$

Verbal

What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?

The rational function will be represented by a quotient of polynomial functions.

What is the fundamental difference in the graphs of polynomial functions and rational functions?

If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?

The numerator and denominator must have a common factor.

Can a graph of a rational function have no vertical asymptote? If so, how?

Can a graph of a rational function have no x -intercepts? If so, how?

Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator.

Algebraic

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

$f\left(x\right)=\frac{x-1}{x+2}$

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

$f\left(x\right)=\frac{{x}^{2}+4}{{x}^{2}-2x-8}$

$f\left(x\right)=\frac{{x}^{2}+4x-3}{{x}^{4}-5{x}^{2}+4}$

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

$f\left(x\right)=\frac{4}{x-1}$

$f\left(x\right)=\frac{2}{5x+2}$

V.A. at $\text{\hspace{0.17em}}x=–\frac{2}{5};\text{\hspace{0.17em}}$ H.A. at $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals $\text{\hspace{0.17em}}x\ne –\frac{2}{5}$

$f\left(x\right)=\frac{x}{{x}^{2}-9}$

$f\left(x\right)=\frac{x}{{x}^{2}+5x-36}$

V.A. at H.A. at $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals

$f\left(x\right)=\frac{3+x}{{x}^{3}-27}$

$f\left(x\right)=\frac{3x-4}{{x}^{3}-16x}$

V.A. at H.A. at $\text{\hspace{0.17em}}y=0;$ Domain is all reals

$f\left(x\right)=\frac{{x}^{2}-1}{{x}^{3}+9{x}^{2}+14x}$

$f\left(x\right)=\frac{x+5}{{x}^{2}-25}$

V.A. at $\text{\hspace{0.17em}}x=-5;\text{\hspace{0.17em}}$ H.A. at $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals $\text{\hspace{0.17em}}x\ne 5,-5$

$f\left(x\right)=\frac{x-4}{x-6}$

$f\left(x\right)=\frac{4-2x}{3x-1}$

V.A. at $\text{\hspace{0.17em}}x=\frac{1}{3};\text{\hspace{0.17em}}$ H.A. at $\text{\hspace{0.17em}}y=-\frac{2}{3};\text{\hspace{0.17em}}$ Domain is all reals $\text{\hspace{0.17em}}x\ne \frac{1}{3}.$

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

$f\left(x\right)=\frac{x+5}{{x}^{2}+4}$

$f\left(x\right)=\frac{x}{{x}^{2}-x}$

none

$f\left(x\right)=\frac{{x}^{2}+8x+7}{{x}^{2}+11x+30}$

$f\left(x\right)=\frac{{x}^{2}+x+6}{{x}^{2}-10x+24}$

$f\left(x\right)=\frac{94-2{x}^{2}}{3{x}^{2}-12}$

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

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