combining and composite functions

like numbers, functions can be added, subtracted, multiplied, and divided (except where the denominator is zero) to produce new functions. If f and g are functions, then for every x that belongs to the domains of both f and g (that is, for x e D(f) N D(g)), we define functions f + g, f — g, and fg by the formulas

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

$$At-any-point-of(D(f)capD(g))-at-which(g(x)\ne 0),we,can,also,define,the,function(\frac{f}{g})bytheformula[\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}\text{(where }g(x)\ne 0\text{)}.]$$

Composite Functions

Composition is another method for combining functions.

DEFINITION If f and g are functions, the composite function f°g (“f composed with g”) is defined by $(f\circ g)(x)=f(g(x)).$

The definition implies that f ° g can be formed when the range of g lies in the domain of f. To find (f ° g)(x), first find g(x) and second find f(g(x)). Figure 1.27 pictures (f ° g) as a machine diagram, which shows the composite as an arrow diagram.

this is possible like the following example: