# 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) â‰ 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 (- âˆž , 7] and that of g is [5, âˆž ), 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 â‰¤ cos x â‰¤ 1 for each x Îµ

**R**, is defined for each x Îµ

**R**.) Thus, the domain of f + g, f - g and f

^{.}g is

**R**. Also, since 2 - cos x > 0 for each x Îµ

**R**, the domain of is also

**R**.

Sometimes we will write f

^{2}to denote f

^{.}f. For instance, if f(x) = 8x, then

f

^{2}(x) = (f

^{.}f) (x) = f(x)

^{.}f(x) = (8x) (8x) = 64x

^{2}.

Similarly, we denote f

^{2.}f by f

^{3}, f

^{3}

^{.}f by f

^{4}, and so on.