Algebra of real functions

Let f and g be real functions with the same domain X ⊂ R. Their sum and difference work value by value: (f + g)(x) = f(x) + g(x) and (f − g)(x) = f(x) − g(x) at every x in X.

For a real number α (a scalar), the function αf is defined by (αf)(x) = α·f(x), x ∈ X.

The product is (fg)(x) = f(x)·g(x), x ∈ X; this is called pointwise multiplication.

The quotient is (f/g)(x) = f(x)/g(x), defined only where g(x) ≠ 0. Its domain is X with the zeros of g removed.

Example: with f(x) = 3x − 2 and g(x) = x + 1 on R, (f + g)(x) = 4x − 1, (f − g)(x) = 2x − 3, (fg)(x) = 3x² + x − 2, (2f)(x) = 6x − 4, and (f/g)(x) = (3x − 2)/(x + 1) for x ≠ −1.

When f and g have different natural domains, the combined function lives on the common part. For f(x) = √x and g(x) = x − 4, f + g is defined on [0, ∞), and f/g on [0, ∞) − {4}.