sum and product and quotient of
functions∗
pahio†
2013-03-22 0:14:16
Let A be a set and M a left R-module. If f : A → M and g : A → M ,
then one may define the sum of functions f and g as the following function
f +g : A → M :
(f +g)(x) := f (x)+g(x) ∀x ∈ A
If r is any element of the ring R, then the scalar-multiplied function rf : A → M
is defined as
(rf )(x) := r·f (x) ∀x ∈ A.
Let A again be a set and K a field or a skew field. If f : A → K and
g : A → K, then one can define the product of functions f and g as the function
f g : A → K as follows:
(f g)(x) := f (x)·g(x) ∀x ∈ A
The quotient of functions f and g is the function
defined as
f (x)
f
(x) :=
g
g(x)
.
f
: {a ∈ A .. g(a) 6= 0} → K
g
.
∀x ∈ Ar{a ∈ A .. g(a) = 0}.
(x)
In particular, the incremental quotient of functions f (y)−f
, as y tends to
y−x
x, gave rise to the important concept of derivative. As another example, we can
with a clear conscience say that the tangent function is the quotient of the sine
and the cosine functions.
∗ hSumAndProductAndQuotientOfFunctionsi
created: h2013-03-2i by: hpahioi version:
h40190i Privacy setting: h1i hDefinitioni h03E20i
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You can reuse this document or portions thereof only if you do so under terms that are
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