1. Which of the following definitions are functions and which are relations:

a. $f(x) = x + 2$ = Function
b. $g(x) = x - 2$ = Function
c. $h(x) = x * 2$ = Function
d. $i(x) = x / 2$ = Function

2. Which of the following definitions are functions and which are relations:

a. $j(x) = x + x$ = Function Same as x * 2.
b. $k(x) = x - x$ = Relation
c. $l(x) = x * x$ = Relation
d. $m(x) = x / x$ = Relation

3. Can you write descriptions of the functions in problems 1 and 2? What is the difference?

The functions in exercise 1 use a variable and a constant. The function and relationships in exercise 2 depend on the value of the variable x. In exercise 1 the functions are guaranteed to be unique because x is changed in a consistent way. In question 2 the last three functions are relations because the function value can re-occur. K(x) is a relation because for any value of x the answer will always be 0, l(x) is a relation because it has the the same values for positive and negative values of x, and m(x) has two values: either 1 or -1 and is undefined at 0. When we evaluate the relationships of the variables in terms of themselves we begin to explore the functionality of the operators. $x + x$ shows the relationship between addition and multiplication. $x * x$ shows a similar relationship between multiplication and exponentiation. (Is $n(x) = x * x * x$ a function or a relation?) $x - x$ and $x / x$ show the identity property of subtraction and division respectively. (What are the identity properties of addition and multiplication? Are these functions or relationships?)