FUNCTIONS

A function is essentially a recipe, a roadmap, an instruction booklet, an algorithm. In other words, take this something-or-other and do such and such to it. Clearly, what you come out with depends on what goes in. Certainly, if I scramble two scoops of ice cream and then fry the scrambled result, I will get an ultimate outcome (melted ice cream?) that is certainly different from what I get if I had started with two regulation chicken eggs.

We use the notation f( ), pronounced eff of, to indicate that we have a function.

Frequently we have f(x), and frequently y = f(x). This indicates that the ultimate value of the output y will depend on the value of the input x.

Example 1:

y = 2 x + 4

This function, that can be written as f(x) = 2 x + 4, is a set of instructions which directs us to take something (in this case x), double it, and add 4.

Thus, if x = 3, y (which as a function of x depends for its value on the value assigned to x) would equal 10 (2 times 3 plus 4). If x = 1, y would equal 6 (2 times 1 plus 4).

When we have more than one function and do not wish to confuse matters by overly repeating f(x), we may distinguish the functions with subscripts such as fl(x), f2(x), f3(x) or with other letters such as f(x), g(x), h(x).

NON-FUNCTIONS

In order to be a function, the set of directions must be completely reliable. That is, every time we start with eggs, we must come out with the exact same omelet. Or, if we start with x = 2, we must always come out with the exact same value for y.

When graphed, this can be verified using the vertical line test. If there is any value of x where you can draw a vertical line and hit the graphed expression at more than one spot, then the expression is not a function.

Example 2:

x2 + y2 = 9

We can solve this for y to obtain y = ± Ö (9-x2). Although this y certainly depends for its value on the value(s) selected for x, if x = 0, we would obtain y = ± 3, two possible answers. Thus the recipe is unreliable - we put in x = 0 and sometimes we get +3 and sometimes we get - 3. Thus, although y = ± Ö (9-x2), y is not a function and it cannot be written as f(x) = ± Ö (9-x2).

Of course, we recognize x2 + y2 = 9 as the generator of a circle of radius 3 centered at the origin (0,0), and at all values - 3 £ x £ +3 the graph would fail the vertical line test.

EVALUATING FUNCTIONS

In example 1 above we evaluated f( x ) = 2 x + 4 when x = 3 and when x = 1. Notationally we can ask for this evaluation in simpler form. We use the eff of notation to express the directions, and we typically use x as the for-instance insertion. We could say "take something, double it, and add 4" or we could simply direct your actions by using f( x ) = 2 x + 4 to say "well, if it was x, then we would come out with 4 more than x doubled."

Thus, f( 3 ) directs us to "take 3, double it, and add 4", and f( 1 ) directs us to "take 1, double it, and add 4".

Consistently, then, we could direct f( t ) which tells us to "take t, double it, and add 4", or we could have f(2x) which tells us to "take 2x, double it, and add 4" (which would, of course, result in y = f(2x) = 4x + 4.

COMBINATION OF FUNCTIONS

Functions can be combined using the ordinary arithmetic operations just as we can combine the algebraic expressions which the function notation represents. That is, given an f(x) and a g(x), we can add, subtract, multiply and divide them by adding, subtracting, multiplying or dividing their algebraic expressions. Please notice the use of notation such as (f + g)(x), (f - g)(x), (f · g)(x), and (f / g)(x).

Example 3:

f(x) = 3x - 5

g(x) = x2 + 2x - 9

Addition: (f + g)(x) = f(x) + g(x) = (3x - 5) + ( x2 + 2x - 9)

= x2 + 5x - 14

Subtraction: (f - g)(x) = f(x) - g(x) = (3x - 5) - ( x2 + 2x - 9)

= 3x - 5 - x2 - 2x - -9

= -x2 + x + 4

Multiplication: (f · g)(x) = f(x) · g(x) = (3x - 5) ( x2 + 2x - 9)

= 3x3 + 6x2 - 27x - 5x2 - 10x + 45

= 3x3 + x2 - 37x + 45

Division: (f / g)(x) = f(x) / g(x) = (3x - 5) ¸ ( x2 + 2x - 9)

COMPOSITION OF FUNCTIONS>

The concept of the composition of functions is fairly similar to some of the comments made above in the section on Evaluating Functions. Repeating part of that: given f( x ) = 2 x + 4, then f(2x) = 2 (2x) + 4 = 4x + 4.

To put it another way, [Example 4] assume that f( x ) = 2 x + 4 and that g( x ) = 2 x. We use notation (f o g)(x), sometimes pronounced "fog of x", to represent the nested concept f (g(x)), spoken as eff of gee of x. The direction is to take the rule laid out by f(x) and to insert or plug in g(x) wherever we had x.

Thus, (f o g)(x) = f(g(x)), in this example, would be 2 g(x) + 4, but since g( x ) = 2 x, then:

(f o g)(x) = f(g(x)) = 2 g(x) + 4 = 2(2x) + 4 = 4x + 4.

Ó Copyright 2001, Mark Wiener.