082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.4.2 Rationalfunctions
2.4 Review of some common engineering functions and techniques 75
Arational function,R(x), has the form
R(x) = P(x)
Q(x)
where P and Q are polynomial functions; P is the numerator and Q is the
denominator.
The functions
R 1
(x) = x +6
x 2 +1
R 2
(t) = t3 −1
2t+3
R 3
(z) = 2z2 +z−1
z 2 +3z−2
areallrational.Whensketchingthegraphofarationalfunction,y = f (x),itisusualto
drawupatableofxandyvalues.Indeedthishasbeencommonpracticewhensketching
anygraphalthoughtheuseofgraphicscalculatorsisnowreplacingthiscustom.Itisstill
usefultoanswer questions such as:
‘Howdoesthefunctionbehaveasxbecomeslargepositively?’
‘Howdoes thefunctionbehaveasxbecomeslarge negatively?’
‘Whatis thevalueof thefunctionwhenx = 0?’
‘At what valuesofxis thedenominator zero?’
Figure 2.21 shows a graph of the function y = 1+2x = 1 + 2. As x increases, the
x x
value ofyapproaches 2.We write thisas
y→2 as x→∞
and say ‘ytends to2asxtends toinfinity’.Also fromFigure 2.21, we see that
y→±∞ as x→0
Asx → ∞,thegraphgetsnearerandnearertothestraightliney = 2.Wesaythaty = 2
is an asymptote of the graph. Similarly,x = 0, that is theyaxis, is an asymptote since
the graph approaches the linex = 0 asx → 0.
If the graph of any function gets closer and closer to a straight line then that line is
calledanasymptote.Figure2.22illustratessomerationalfunctionswiththeirasymptotes
indicatedbydashedlines.InFigure2.22(a)theasymptotesarethehorizontalliney = 3
y
2
x
Figure2.21
Thefunction:y = 1+2x = 1 x x +2.