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

56 Chapter 2 Engineering functions
–6 –4 –1 0 2 3 4
Figure2.2
The intervals (−6,−4),[−1,2], (3,4]depicted onthe real line.
real line. The real line extends indefinitely to the left and to the right so that any real
number can be represented.
Sometimesweareinterestedinonlyasmallsection,orinterval,oftherealline.We
write [1, 3] to denote all the real numbers between 1 and 3 inclusive, that is 1 and 3 are
includedintheinterval.Thustheinterval[1,3]consistsofallrealnumbersx,suchthat
1 x3. The square brackets, [ ], are used to denote that the end-points are included
in the interval and such an interval is said to be closed. The interval (1, 3) consists of
all real numbers x, such that 1 < x < 3. In this case the end-points are not included
and the interval is said to be open. Brackets, (), denote open intervals. An interval may
be open at one end and closed at the other. For example, (1, 3] is open at the left and
closed at the right. It consists of all real numbersx, such that 1 < x 3, and is known
as a semi-open interval. Open and closed intervals can be represented on the real line.
A closed end-point is denoted by •; an open end-point is denoted by ◦. The intervals
(−6,−4), [−1,2] and (3, 4] are illustratedinFigure 2.2.
An upper bound of a set of numbers is any number which is greater than or equal
toeverynumberinthegivenset.So,forexample,7isanupperboundfortheset[3,6].
Clearly, 7 isgreater than every number inthe interval[3, 6].
Alowerboundofasetofnumbersisanynumberwhichislessthanorequaltoevery
number inthe given set.For example, 3 isalower bound forthe set (3.7,5).
Note that upper and lower bounds are not unique. Both 3 and 10 are upper bounds
for (1, 2). Both −1 and −3 are lower bounds for[0, 6].
TechnicalcomputinglanguagessuchasMATLAB ® usuallyhavefunctionsthatautomaticallygenerateasetofnumberswithinaparticularinterval.InMATLAB
® wecould
generate a set of time values,t, by typing:
t= 0:0.1:1
Thisgeneratesasetofrealnumbersfromtheinterval[0,1]storedinarowvectort,each
individual number being separated by an increment of 0.1. The values oft generated
are:
0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000
0.8000 0.9000 1.0000
2.3 BASICCONCEPTSOFFUNCTIONS
Loosely speaking, we can think of a function as a rule which, when given an input,
producesasingleoutput.Ifmorethanoneoutputisproduced,theruleisnotafunction.
Considerthefunctiongivenbytherule:‘doubletheinput’.If3istheinputthen6isthe
output. Ifxisthe input then 2x isthe output, as shown inFigure 2.3.
If the doubling function has the symbol f wewrite
f:x→2x
or more compactly,
f(x)=2x