Understanding Engineering Mathematics

m,nare ‘constant’ and to think in terms of apower functionxn, etc. where the base,x,
is variable andnis given. However, it is just as possible to let the base be fixed and let
the power be any variable. This gives a function of the form

f(x)=ax a= 0

where thebasea is now regarded as constant, and theexponentx can vary. Such a
function is calledexponential. But of coursexis still just an index, and obeys all the
usual rules of indices (70

➤
). One important point to note at this stage is that sincex
can takefractionalvalues it is essential thatabepositiveto deliver real values of the
function for all values ofx.xitself can of course be positive or negative. The results of
Section 4.2.1 give us a feel for the shape of the graph of exponential functions. Basically
such graphs can take three different forms:

• basea>1, the graph increases steadily from left to right. We sayax
  increases monotonicallywithx;


• a<1 the graph decreases steadily from left to right−axdecreases
  monotonicallywithx;


• ifa=1 we get the straight liney=1.

These results are illustrated in the graphs of Figure 4.2.

y

0 x

1

y (^) =
(^) ax
a < 1
y^
=^ a
x
a > 1
y = ax
a = 1
Figure 4.2The exponential functionsax,a−x.
The exponential function satisfies all the usual laws of indices, which are worth repeating
here in the new notation:
axay=ax+y
ax
ay
=ax−y
(ax)y=axy
(ab)x=axbx
a−x=
1
ax