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2.3 Basic concepts of functions 61
both of which have the same result:
ans=9
affirming that f (x) =x 2 isamany-to-one function.
Noticethatthevariableusedbythefunctionisdefinedinbracketsafterthe@sign.This
indicatesthattheinputtothefunctionisxandthecommandcreatesafunctionhandle,
f. Giving the function a handle enables ittobe used elsewhere inthe program.
More complicated functions are usually created in a separate file and saved on the
computer’s internal storage devices. They can be easily reused to create sophisticated
programs. In MATLAB ® these files are saved with the file name extension .m and are
often termedm-files.
2.3.5 Parametricdefinitionofafunction
Functions are often expressed in the formy(x). For every value ofxthe corresponding
valueofycanbefoundandthepointwithcoordinates (x,y)canthenbeplotted.Sometimes
it is useful to expressxandycoordinates in terms of a third variable known as a
parameter.Commonlyweuset or θ todenoteaparameter.Thusthecoordinates (x,y)
of the points on a curve can be expressed inthe form
x=f(t)
y=g(t)
For example, given the parametric equations
x =t 2 y=2t 0t5
we can calculatexandyfor various values of the parametert. Plotting the points (x,y)
produces partof a curve known as a parabola.
2.3.6 Compositionoffunctions
Considerthefunctiony(x) = 2x 2 .Wecanthinkofy(x)asbeingcomposedoftwofunctions.Onefunctionisdescribedbytherule:‘squaretheinput’,whiletheotherfunction
isdescribed by the rule:‘double the input’. This isshown inFigure 2.9.
y(x)
8
x
g
square
the input
Figure2.9
Thefunction:y(x) =h(g(x)).
h
x 2 double
2x
the input
2
2
–2 1 x
Mathematically, ifh(x) = 2x andg(x) =x 2 then
y(x) = 2x 2 = 2(g(x)) =h(g(x))
The form h(g(x)) is known as a composition of the functions h and g. Note that the
compositionh(g(x)) isdifferent fromg(h(x)) as Example 2.5 illustrates.