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Sum, Difference, Product and Quotients of Functions


Just as real numbers can be added, subtracted, multiplied and divided to produce other numbers, there is a useful way of adding, subtracting, multiplying and dividing functions to produce other functions. We define these operations as follows:

Given functions f and g, their sum f + g, difference f - g, product f . g and quotient are defined by
(f + g)(x) = f(x) + g(x),
(f - g)(x) = f(x) - g(x)
(f . g)(x) = f(x) . g(x),
  provided g(x) 0.

For the functions f + g, f - g and f. g, the domain is defined to be the intersections of the domains of f and g, and for the domain is the intersection with the points where g(x) = 0 is excluded.

Illustration
Let f and g be the functions f(x) = and g(x) = Then the formulae for f + g, f - g, f. g and f/g are
(f + g) (x) = f(x) + g(x) = ,
(f - g) (x) = f(x) - g(x) =
(f . g) (x) = f(x) . g(x) =

Since the domain of f is (- , 7] and that of g is [5, ), the domain of f + g, f - g and f .g is the interval [5, 7]. Since g(x) = 0 when x = 5, we must exclude this point to obtain the domain of as (5, 7].

Illustration
Let f and g be the functions f(x) = sin x and g(x) = Then the formulae for f + g, f - g, f . g and are
(f + g)(x) = f(x) +g(x) = sin x +
(f - g)(x) = f(x) - g(x) = sin x -
(f . g)(x) = f(x) . g(x) = sin x ,


The domain of f is R and that of g is also R. (As - 1 cos x 1 for each x ε R, is defined for each x ε R.) Thus, the domain of f + g, f - g and f . g is
R. Also, since 2 - cos x > 0 for each x ε R, the domain of is also R.
Sometimes we will write f2 to denote f . f. For instance, if f(x) = 8x, then
f2 (x) = (f . f) (x) = f(x) . f(x) = (8x) (8x) = 64x2.
Similarly, we denote f2. f by f3, f3. f by f4, and so on. 




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