Math Help

In a Nut Shell: Functions of one (or more) independent variables can be combined in

various ways to develop new functions. In each of the simple cases below the single

independent variable is x.

(cf)(x) = c f(x) constant, c, times a function, f(x)

(f + g)(x) = f(x) + g(x) addition of functions f(x) and g(x)

(f – g)(x) = f(x) – g(x) subtraction of g(x) from f(x)

(f * g)(x) = f(x) * g(x) where * denotes multiplication of f(x) times g(x)

(f / g)(x) = f(x) / g(x) division of f(x) / g(x) where g(x) ≠ 0

Here are four simple examples using f(x) = 3x + 5 and g(x) = √6x

(f + g)(x) = 3x + 5 + √6x obtain new function by addition

(f – g)(x) = 3x + 5 - √6x obtain new function by subtraction

(f * g)(x) = (3x + 5) (√6x) obtain new function by multiplication

(f/g)(x) = (3x + 5) / (√6x) obtain new function by division

Another function, called the composite function, can be formed by replacing the

independent variable, x, in one function with the other function. For example:

Let f(x) and g(x) be defined as follows: f(x) = 3x + 5, g(x) = √6x

The composition of f and g is expressed as f( g(x) ), reads as f of g(x).

That is replace x with g(x). In this case f(g(x) ) = 3(√6x) + 5

Likewise the composition of g and f is g( f(x) ) , reads as g of f(x).

That is replace x with f(x). In this case g(f(x) ) = √ [ 6(3x + 5) ]

Note that this type of calculation will be used later for the chain rule of differentiation

of functions. The chain rule is very important.