Algebraic Operations on Functions

Significance

In calculus we work with functions whose domains and codomains (ranges) are subsets of the real numbers. This affords us the luxury of being able to extend the following familiar operations from the real numbers to the functions in this Field Guide:

  • Addition, Subtraction, Mutliplication, and Division.

  • Forming negations and reciprocals.

  • Taking powers and extracting roots.

For an applicaton, see the discussion of algebraic functions in the section on general types of functions.


Standard Notation

Given two functions f(x) and g(x) whose domains are ranges are subsets of the real numbers, we define (f + g)(x) to be the function whose value at x is the sum of f(x) and g(x). Symbolically,

(f + g)(x) = f(x) + g(x).

The domain of f+g is the intersection of the domains of f and g since to compute (f + g)(x) it is necessary and sufficient to compute both f(x) and g(x).

Other operations on functions are defined similarly:

  • (fg)(x) = f(x)g(x)

  • (f/g)(x) = f(x)/g(x) [if g(x) = 0, then x is not in the domain of (f/g)].

  • fp(x) = (f(x))p for any real exponent p with the domain of fp consisting of those points for which the p-th power of f(x) makes sense.
Example: if f(x) = 3 sin (5 x) and g(x) = x2, then

(f+g)(x) =3 sin(5 x) + x2
(fg)(x) =3 sin(5 x)*x2
(f-g)(x) =3 sin(5 x) - x2
(f/g)(x) =3 sin(5 x) / x2
The domains of both f and g are all real numbers, but the domain of f/g is { x | x 0}.

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