Cambridge International AS and A Level Mathematics Pure Mathematics 1

Composite functions

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4

In this case the composite function would be (to the nearest degree)

C  34°C too cold

35°C  C  40°C all right

C  41°C too hot.

In algebraic terms, a composite function is constructed as

Input x 

f

Output f(x)

Input f(x) 

g

Output g[f(x)] (or gf(x)).

Thus the composite function gf(x) should be performed from right to left: start
with x then apply f and then g.

Notation

To indicate that f is being applied twice in succession, you could write ff(x) but
you would usually use f^2 (x) instead. Similarly g^3 (x) means three applications of g.

In order to apply a function repeatedly its range must be completely contained
within its domain.

Order of functions

If f is the rule ‘square the input value’ and g is the rule ‘add 1’, then

x 

f

x^2 

g
x^2 + 1.
square add 1

So gf(x) = x^2 + 1.

Notice that gf(x) is not the same as fg(x), since for fg(x) you must apply g first. In
the example above, this would give:

x 

g

(x + 1) 

f
(x + 1)^2
add 1 square
and so fg(x) = (x + 1)^2.

Clearly this is not the same result.

Figure 4.4 illustrates the relationship between the domains and ranges of the
functions f and g, and the range of the composite function gf.

I g

GRPain RI I Uange RI I Uange RI gI

GRPain RI g

gI

Figure 4.4

Read this as

‘g of f of x’.