Understanding Engineering Mathematics

Polynomial:f(x)=anxn+an− 1 xn−^1 +···+a 1 x+a 0 (allx)

Reciprocal: f(x)=

1

x

x= 0

Rational function – for example:

f(x)=

x+ 1

(x− 1 )(x+ 2 )

x= 1 ,− 2

Here the coefficients,a,b,c,d,an,...etc. are constants, or more strictly independent
ofx. Such algebraic functions occur frequently in mechanics and electrical circuits, for
example. Rational functions are particularly important in control theory, where the roots
of the denominators (the so-calledpoles) have physical meaning for the stability of a
system.
The other standard functions – exponential and trigonometric – will be studied later.
They all reflect some important characteristic physical behaviour – for example the
exponential function represents the ‘law of natural growth’ whereby a quantity
increases at a rate proportional to the amount present. The trigonometric functions
sinxand cosinexdescribe wavelike behaviour such as occurs in an oscillating electrical
circuit.
The formy=f(x)defines anexplicit formof a function ofx. It is sometimes conve-
nient to define a functionimplicitly. For example the reciprocaly= 1 /xmay be defined
implicitly asxy=1.
Another form of representation of a function is by means of a parameter in terms
of which bothxandyare expressed – this is called aparametric representation.For
example the position (x,y) of a projectile in a plane at timetmay be expressed by
x= 3 + 2 t,y= 4 + 2 t−t^2.

Solution to review question 3.1.1

Iff(x)=

x+ 1

x^2 + 2

then

(i) f( 0 )=

0 + 1

02 + 2

=

1

2

(ii) f(− 1 )=

− 1 + 1

(− 1 )^2 + 1

= 0

3.2.2 Plotting the graph of a function

➤

88 108➤

We can learn a lot about a functiony=f(x)from itsgraphplotted againstx-,y-axes,
in a planeCartesian coordinate system. We choose an appropriate sequence of values
ofxand calculate the corresponding values ofyand plot the resultingx,ycoordinates
against the axes. We may then be able to draw a continuous curve through the points.
Figure 3.1 illustrates this for the straight line graph of the linear functiony= 2 x+1.
For this we only need two points, say (0, 1) and (1, 3), to define the resulting straight
line graph.