Rational Functions: Find function having listed properties.

[adonai]

Write the equation of one rational function f(x) = p(x)/q(x) having the indicated properties, in which the degrees of p and q are as small as possible.
a) f has a vertical asymptote given by x=1
b) f has a slant asymptote whose equation is y=x
c) f has a y-intercept at 2
d) f has x-intercepts at -1 and 2 :?:

 

[soroban]

Re: Rational Functions

Hello, adonai!

\(\displaystyle \text{Write the equation of one rational function: }\:f(x) \:=\:\frac{p(x)}{q(x)},\:\text{ having the indicated properties,}\)
\(\displaystyle \text{in which the degrees of }p(x)\text{ and }q(x)\text{ are as small as possible.}\)

. . \(\displaystyle \text{a) }f(x)\text{ has a vertical asymptote: }\:x = 1\)
. . \(\displaystyle \text{b) }f(x)\text{ has a slant asymptote: }\:y = x\)
. . \(\displaystyle \text{c) }f(x)\text{ has y-intercept: }\,(0,2)\)
. . \(\displaystyle \text{d) }f(x)\text{ has x-intercepts: }\,(-1,0)\text{ and }(2,0).\)

\(\displaystyle \text{From (d), the numerator has: }\x+1)(x-2)\)

\(\displaystyle \text{From (a), the denominator has: }\:x -1\)

\(\displaystyle \text{The function (so far) is: }\:f(x) \;=\;\frac{(x+1)(x-2)}{x-1}\)


\(\displaystyle \text{We find that: }\:f(0) \:=\:\frac{(0+1)(0-2)}{0-1} \:=\:2\quad\hdots\quad\text{The y-intercept is 2}\)


\(\displaystyle \text{Conside the limit as }x\to\infty\!:\quad \lim_{x\to\infty}\frac{x^2-x-2}{x-1} \;=\;\lim_{x\to\infty}\left(x - \frac{2}{x-1}\right)\)

\(\displaystyle \text{Divide top and bottom by }x\!:\quad\lim_{x\to\infty}\frac{x - 1 - \frac{2}{x}}{1-\frac{1}{x}} \;=\;x\quad\hdots\quad\text{ The slant asymptote is: }y = x\)


\(\displaystyle \text{Therefore: }\;\boxed{f(x) \;=\;\frac{(x+1)(x-2)}{x-1}}\)