Math Analysis – Precalculus, Sullivan 10th Edition
Ch5Sec1 1 12/1/2019
Section 5.1 – Composite Functions
From Section 2.1, you know functions, like numbers, can be added, subtracted, multiplied, and divided.
If f and g are functions:
Their sum,
,gf
is the function defined by
).x(g)x(f)x)(gf(
The domain of
)x)(gf(
consists of the numbers x that are in the domain of f and in the domain of g.
Their difference,
,gf
is the function defined by
).x(g)x(f)x)(gf(
The domain of
)x)(gf(
consists of the numbers x that are in the domain of f and in the domain of g.
Their product,
,gf
is the function defined by
).x(g)x(f)x)(gf(
The domain of
)x)(gf(
consists of the numbers x that are in the domain of f and in the domain of g.
Their quotient,
,
g
f
is the function defined by
.0)x(gwhere,
)x(g
)x(f
)x(
g
f
The domain of
)x(
g
f
consists of the numbers x for which
0)x(g
that are in the domain of f and
in the domain of g.
Example 1: Let f and g be two functions defined as
2
x5)x(f
and
.1x3)x(g
Find all four operations
of f and g and the domain for each operation.
a)
)x(g)x(f)x)(gf(
b)
)x(g)x(f)x)(gf(
1x3x5 2
)1x3(x5 2
Domain: All real numbers
1x3x5 2
Domain: All real numbers
c)
)x(g)x(f)x)(gf(
d)
)x(g
)x(f
)x(
g
f
1x3
x5 2
Domain: All real numbers except
3
1
x
OR
3
1
xx
Form a Composite Function
Given two functions f and g, the composite function, denoted by
gf
(read as “f composed with g”), is defined by
)x(g)x(f)x()gf(
))x(g(f
.
The domain of
)x()gf(
is the set of all numbers x in the domain of g such that
)x(g
is in the domain of f.
Look at Figure 2. Only those values of x in the domain of g for which g(x) is in the domain of f can be in the domain of
gf
. The reason is that if g(x) is not in the domain of f, then f(g(x)) is not defined. Because of this, the domain of
gf
is a subset of the domain of g; the range of
gf
is a subset of the range of f.
)1x3(x5 2
23 x5x15
Domain: All real numbers
