Consider the rational function .

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

Consider the rational function

p(x)=\dfrac{1}{x-1}+2

. (i) Find the equations of all vertical and horizontal asymptotes of the graph of

y=p(x)

. (ii) Determine the coordinates of the point(s) where the graph intersects the coordinate axes. (iii) Analyse the behaviour of

p(x)

x\to1^{-}

x\to1^{+}

, and as

x\to\pm\infty

. (iv) Using the results of parts (i)–(iii), sketch the graph of

y=p(x)

for

-4\le x\le 6

. Draw the asymptotes as dashed lines and clearly mark all intercepts.