Example

Let the functions f : x → x² and g : x → x³. Determines the functions that are defined as

A.                  f + g,

B.                  f - g,

C.                  fg,

D.                 (2f + 3g), and

E.                  f  /g.

Solution

A.                  (f + g) (x) = x² + x³

B.                  (f - g) (x) = x² - x³

C.                  (fg) (x) = x⁵

D.                 (2f + 3g) (x) = 2f (x) + 3g (x)

                                     = 2f (x) + 3g (x)

                                     = 2x² + 3x³.

(v)        Now, g (x) = 0 ⇔ x² = 0 ⇔ x = 0. Thus, to define the function f /g, we will assume just non-zero values of x. If x ≠ 0,

        . Thus, f /g is the function f /g: x → 1/x, when x≠0.