Understanding Engineering Mathematics

3.2.5 Composition of functions

➤

89 108➤

Often we need to combine functions. For example in differentiating a function such as
1
x^2 + 1

it is useful to regard it as a function(the reciprocal)of a function(x^2 +1)

(Chapter 8). More specifically, iff(x)andg(x)are functions with appropriate domains
and ranges we can define theircompositionf(g(x))which is the result of evaluatingfat
the values given byg(x)for each value ofx. Note thatf(g(x))andg(f(x))are different
in general.

Example

f(x)=x^2 + 1 ,g(x)=

1

x

.

f(g(x))=

(

1

x

) 2

+ 1 =

1

x^2

+ 1

while

g(f(x))=

1

f(x)

=

1

x^2 + 1

Solution to review question 3.1.5

(i)f(g(x))=g(x)+ 1 =

1

x− 1

+ 1 =

x

x− 1

.

(ii) g(f(x))=

1

f(x)− 1

=

1

x+ 1 − 1

=

1

x

.

Remember to tidy up your result at the end.

3.2.6 Inequalities

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89 109➤

We introduced inequalities in Section 1.2.2. Their basic properties are the following:

• Ifa<bandc<dthena+c<b+d

That is, ifais less thanbandcis less thandthen the sum ofaandcwill be less
than the sum ofbandd. This is a fairly obvious property of numbers in general, whether
positive or negative.

• Ifadthena−c<b−d

Perhaps the easiest way to see this is to plot the numbersa,b,c,dsatisfying the above
conditions, on the real line. Then,a−cis the ‘distance’ betweenaandcandb−dis
the ‘distance’ betweenbandd, and it is then pictorially obvious that the former is less
than the latter. Or, sincea<bwe can writea=b−pwherepis a positive number
and we can similarly writec=d+qwhereqis another positive number. Thena−c=
b−p−(d+q)=b−d−(p+q)from which, sincep+qis positive, it follows that
a−c<b−d, as required.

• Ifa<bandb<cthena<c