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(Section 1.6: Combining Functions)
1.6.5
PART C: LINEAR COMBINATIONS OF FUNCTIONS
Let
f
and
g
be functions. Let
c
and
d
be real numbers, possibly 0.
cf
+
dg
is then called a linear combination
of
f
and
g
.
• Its domain is the overlap (intersection)
Dom
f
()
±
Dom
g
.
• More generally, a linear combination of objects is a sum of constant
multiples of those objects.
Example 4 (Linear Combinations)
a)
3
f
+
4
g
,
1
2
f
±
g
,
±
g
, and 0 are linear combinations of
f
and
g
.
b)
2
f
+
3
g
±
4
h
and
±
3.7
f
+
g
are linear combinations of
f
,
g
, and
h
.
§
• In Section 1.7, we will see that a linear combination of
even
functions is
even
.
The same goes for
odd
functions.
•
WARNING 4
:
People often erroneously attempt to apply linearity properties in
precalculus. For example, remember that the “square root of a sum” is typically
not
equal to the “sum of the square roots.” Think:
+
4
±
+
4
.
•
In calculus
, we will see important linearity theorems
for limits, derivatives, and integrals.
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 Spring '09
 Real Numbers

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