Section 1-5-1

Chapter 1
Relationships

1.5 The Algebra of Functions

1.5.1 Sums, Products, Differences,
Quotients of Functions

The function descriptions at the end of the preceding section suggest some possibilities for building functions from other functions. Indeed, there is a natural way to apply the operations of algebra (addition, subtraction, multiplication, division, extraction of roots, absolute value, and so on) to function-objects to produce new functions.

Example 1

We can describe the function f ( x ) = x 3 + 4 ⁢ x [see part (a) of Checkpoint 1 in Section 1.4] as the sum of the cubing function c ( x ) = x 3 and the “multiply by 4” function m ( x ) = 4 ⁢ x . In Figure 1 we show the graphs of `c`, `m`, and `f`. Note that `y`-coordinates on the graph of `f` are sums of the corresponding `y`-coordinates on the graphs of `c` and `m`.

Figure 1 The function f(x) as the sum of simpler functions

Definitions The function s defined by s ( x ) = f ( x ) + g ( x ) is the sum of the functions f ( x ) and g ( x ). The function p defined by p ( x ) = f ( x ) g ( x ) is the product of the functions f ( x ) and g ( x ). The function d defined by d ( x ) = f ( x ) - g ( x ) is the difference of the functions f ( x ) and g ( x ). The function q defined by q ( x ) = f ( x ) / g ( x ) is the quotient of the functions f ( x ) and g ( x ). Similar definitions apply for other operations on functions, such as negative, square root, and absolute value.

For example, the functions described in Checkpoint 2 in Section 1.4 are sums and products of the (unspecified) functions `f` and `g`.

Activity 1

Express the function f ( x ) = x 3 + 4 ⁢ x 2 [see Checkpoint 1(b) in Section 1.4] in terms of operations on simpler functions g and h. Is there more than one way to do this? (Use your graphing tool — computer or calculator — to graph `f`, `g`, and `h` as a visual check of your answer.)

Comment on Activity 1

Combining functions with algebraic operations is not just an abstract game — it's a powerful conceptual tool for expressing relationships.

Example 2

Three functions are represented graphically in Figure 2: exports, imports, and trade balance, each as a function of time. The figure is adapted from a government article on Canada's state of trade in 2001. What algebraic operation between functions is represented by this picture?

Figure 2 Canada's exports, imports, and trade balance with
Free Trade Area of the Americas countries
other than U.S.A. and Mexico, 1980-2000

Solution Figure 2 represents graphically the relationship

`text[Exports] -text[Imports]=text[Trade balance.]`

In any given year, this equation states something obvious about subtracting two numbers to get a third. For example,

`text[Exports (]1990text[)] - text[Imports (]1990text[)] = 1.7 - 3.1 = -1.4 = text[Trade balance (]1990text[)]`.

But by applying the subtraction to functions rather than numbers, we can express the entire relationship in Figure 2 by a single, simple formula.

Image credit

Contents for Chapter 1

Chapter 1 Relationships

1.5 The Algebra of Functions

Chapter 1
Relationships