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

64 Chapter 2 Engineering functions
Example2.8 Giveng(x) = x −1 find the inverse ofg.
2
Solution We knowg(x) = x −1 ( ) x −1
2 , and sog−1 =x.Lety= x −1 so that
2 2
But,
and so
g −1 (y) =x
x=2y+1
g −1 (y)=2y+1
Using the same independent variable as forthe functiong, we obtain
g −1 (x)=2x+1
WenotethattheinversesofthefunctionsinExamples2.7and2.8arethemselvesfunctions.
They are called inverse functions. The inverse of f (x) = 2x + 1 is f −1 (x) =
x −1 x −1
, and the inverse ofg(x) = isg −1 (x) = 2x +1. This illustrates the important
point that if f (x) andg(x) are two functions and f (x) is the inverse ofg(x), then
2 2
g(x) is the inverse of f (x). It is important to point out that not all functions possess an
inverse function. Consider f (x) =x 2 , for −∞ <x<∞.
The function, f, is given by the rule: ‘square the input’. Since both a positive and
negative value of x will yield the output x 2 , the inverse rule is given by: ‘take plus or
minus the square root of the input’. As discussed earlier, this is a one-to-many rule and
soisnotafunction.Clearlynotallfunctionshaveaninversefunction.Infact,onlyoneto-onefunctionshaveaninversefunction.Supposewerestrictthedomainof
f (x) =x 2
suchthatx 0.Then f isaone-to-onefunctionandsothereisaninversefunction.The
inverse function is f −1 (x) given by
Clearly,
f −1 (x) = + √ x
f −1 (f(x))=f −1 (x 2 )=x
wherexisthepositivesquarerootofx 2 .Restrictingthedomainofamany-to-onefunction
so that a one-to-one function results is a common technique of ensuring an inverse
function can be found.
2.3.8 Continuousandpiecewisecontinuousfunctions
Wenowintroduceinaninformalwaytheconceptofcontinuousandpiecewisecontinuousfunctions.AmorerigoroustreatmentfollowsinChapter10afterwehavediscussed
limits.Figure2.11showsagraphof f (x) = 1 x .Notethatthereisabreak,ordiscontinuity,
inthe graph atx = 0.The function f (x) = 1 issaidtobe discontinuousatx = 0.
x