Algebra of Functions

Given functions f : D → R and g : D → R, we describe functions f + g, f - g, gf an f/g as follows:

•       f + g : D → R is a function defined by (f + g) (x) = f(x) + g(x)
•       f - g : D → R is a function defined by (f - g)(x) = f(x) - g(x)
•       fg : D → R is a function defined by (fg) (x) = f(x) g(x)
•       f/g : C → R is a function defined by (f/g) (x) = f(x)/g(x) , g(x) ≠ 0,

where C = {x ε D : g(x) ≠ 0}.

Illustration: Let f : [-1, 1] → R and g : R → R be functions defined by f(x) = √(1-x²) and g(x) = x³ + 1. Find the domain of f + g, fg and f/g.

Solution:

   (f+g)(x) = f(x)+g(x) = √(1-x²) + x³ + 1, (f-g)(x) - g(x) = √(1-x²) - x³ = 1,

   (fg)(x) = f(x) g(x) = (x³ + 1) √(1-x²)  and  (f/g)(x) = f(x)/g(x)  = (√(1-x²))/(x³- 1), x ≠ 1.

The domain of each of f + g. f - g and fg is [-1, 1] and of f/g is

                        {x ε [-1, 1] : g(x) ≠ 0} = (-1, 1].

Illustration: Let f(x) = 

Describe the function f/g.

Solution:

We redefine the functions f(x) and g(x) in the intervals as shown below:

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