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Rational Function | $f(x)=\frac{P(x)}{Q(x)}=\frac{{a}_{p}{x}^{p}+{a}_{p-1}{x}^{p-1}+\mathrm{...}+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q-1}{x}^{q-1}+\mathrm{...}+{b}_{1}x+{b}_{0}},Q(x)\ne 0$ |
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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.
For the following exercises, find the domain of the rational functions.
$f(x)=\frac{x-1}{x+2}$
$f(x)=\frac{x+1}{{x}^{2}-1}$
$\text{Allreals}x\ne \u20131,1$
$f(x)=\frac{{x}^{2}+4}{{x}^{2}-2x-8}$
$f(x)=\frac{{x}^{2}+4x-3}{{x}^{4}-5{x}^{2}+4}$
$\text{Allreals}x\ne \u20131,\u20132,1,2$
For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.
$f(x)=\frac{4}{x-1}$
$f\left(x\right)=\frac{2}{5x+2}$
V.A. at $\text{\hspace{0.17em}}x=\u2013\frac{2}{5};\text{\hspace{0.17em}}$ H.A. at $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals $\text{\hspace{0.17em}}x\ne \u2013\frac{2}{5}$
$f(x)=\frac{x}{{x}^{2}-9}$
$f(x)=\frac{x}{{x}^{2}+5x-36}$
V.A. at $\text{\hspace{0.17em}}x=4,\u20139;\text{\hspace{0.17em}}$ H.A. at $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals $\text{\hspace{0.17em}}x\ne 4,\u20139$
$f\left(x\right)=\frac{3+x}{{x}^{3}-27}$
$f(x)=\frac{3x-4}{{x}^{3}-16x}$
V.A. at $\text{\hspace{0.17em}}x=0,4,-4;\text{\hspace{0.17em}}$ H.A. at $\text{\hspace{0.17em}}y=0;$ Domain is all reals $\text{\hspace{0.17em}}x\ne 0,4,\u20134$
$f(x)=\frac{{x}^{2}-1}{{x}^{3}+9{x}^{2}+14x}$
$f(x)=\frac{x+5}{{x}^{2}-25}$
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 $\text{\hspace{0.17em}}x\ne 5,-5$
$f(x)=\frac{x-4}{x-6}$
$f\left(x\right)=\frac{4-2x}{3x-1}$
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.
$f(x)=\frac{x+5}{{x}^{2}+4}$
$f(x)=\frac{{x}^{2}+8x+7}{{x}^{2}+11x+30}$
$f(x)=\frac{{x}^{2}+x+6}{{x}^{2}-10x+24}$
$x\text{-interceptsnone,}y\text{-intercept}\left(0,\frac{1}{4}\right)$
$f(x)=\frac{94-2{x}^{2}}{3{x}^{2}-12}$
For the following exercises, describe the local and end behavior of the functions.
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