Try to remember the board games you know. Do they have any relation with the cartesian

Th hee aaddvvaan ncceedd ssaallvvoo ggaam mee offers a challenge for the expert player. It is played as salvo, except after a salvo of shots is called, the opponent simply announces how many hits were made - but not where or on what ships.

Try to remember the board games you know. Do they have any relation with the cartesian

A. DEFINING A FUNCTION

1. Definition

There are many variables around us. Some of them are closely related to each other and some of them are not. For instance, the number of students in a classroom and the quantity of oxygen in the same classroom are closely related. But the number of students in a classroom and the temperature outside are not related at all. Functions are used to show this relation between variables. Using that relation we can estimate the results for possible cases.

Below, the numbers on the right are related to the numbers on the left:

1  1 2  4 3  9 4  16

You can easily guess the rule that relates the number on the left to the number on the right.

It is: “square the number”. So the relation converts a number x to another number x². We can symbolize this as:

x x² So,

If x = 10, “the square of the number” is 100 (1^stsentence) If x = –4, “the square of the number” is 16 (2^ndsentence) If x = 0.5, “the square of the number” is 0.25 (3^rdsentence)

Obviously, a way of expressing the result of the rule as shown above is not very practical or mathematical. Writing it in the form 10  100 is also not good since it is not clear which rule we are using. That is, the meaning of “” can be confusing. (it may be “add 90 to the number” or “multiply the number by 10” as well). In order to talk about this rule, in our case

“square the number”, we should name it as f. When we apply this rule to x, we get x². So f is the rule that converts x into x². Symbolically,

f(x) = x². Let us rewrite the above sentences once more:

f(10) = 100 (1^stsentence) f(–4) = 16 (2^ndsentence) f(0.5) = 0.25 (3^rdsentence)

We can write each of the numbers above on the “left” and “right” as ordered pairs like (10, 100), (–4, 16), (0.5, 0.25). Note that for each number on the “left”, which is the first component, there corresponds just one number on the “right”, which is the second component. We know that the mathematical relation is a set of ordered pairs. Whenever the first component of an ordered pair is associated with exactly one second component we name that relation a function.

Definition function

A function f is a rule that assigns to each element x in set A exactly one element y or f(x) in set B. Set A is called the domain and set B is called the range of the function f. We name x as the independent vvariable or the argument, and y as the dependent vvariable since the value of y depends on x.

EXAMPLE

17

a. {(0, 2), (0, 3), (1, 6), (2, 4), (3, 5)}

b. {(–3, 1), (–1, –1), (0, 1), (1, 3), (2, –2)}

c. A relation having ordered pairs of the form (radius, area of circle) d. A relation having ordered pairs of the form (name, surname) e. {(1, z), (2, d), (4, f)} with the domain {1, 2, 3, 4}

A function can also be thought of as a set of ordered pairs whose first components are all different. The set of all the first components of the ordered pairs is the domain of the function. The set of all the second components

of the ordered pairs is the range of the function. Consider the set, A = {(cat, dog), (chicken, duck), (cat, mouse)}. The set A would not be a function because the first component, cat, is paired with 2 different second components. Consider the set B = {(1, 2), (2, 6), (3, 9)}. This is a function since each first component has only one second component paired with it. The domain of B is the set {1, 2, 3} and the range of B is the set {2, 6, 9}. C = {(1, 3), (1, 4), (4, 6)}

would not be a function since 1 has two components paired with it, 3 and 4.

B = {(1, 2), (2, 6), (3, 9)}

Differentialis:

If some quantities so depend on other quantities that if the latter are changed the former undergo change, then the former quantities are called functions of the latter.

This denomination is of broadest nature and compromises every method by means of which one quantity could be determined by others. If, therefore, x denotes a variable quantity, then all quantities which depend on x in any way or are determined by it are called functions of it.

Solution a. In order to have a function for each value in the domain we should have exactly one element assigned in the range.

Since 0 is assigned both to 2 and 3, this is not a function.

b. The domain is {–3, –1, 0, 1, 2}. Since there is only one value assigned for each value in the domain, this is a function. Note that in the image on the right –3 and 0 are both assigned to 1, but this does not prevent it from being a function. The elements from the domain may be assigned to the same value from the range. The important point is that

each element of the domain must not have more than one different value from the range.

c. For each given radius there is exactly one possible circle area. This relation is a function.

d. Two people with the same name can have different surnames. This relation is not a function.

e. As in the previous examples if nothing else is stated as the domain, the set of all the first components is the domain. For this example it would be obvious to think that the domain is {1, 2, 4}. This relation is a function if we didn't see the phrase “with domain {1, 2, 3, 4}”.

But the definition of a function states that every element in a domain must be assigned by exactly one element. We can see that the element “3” from the domain remains unassigned. Therefore this relation is not a function.

To each person there corresponds just