# Z x Z functions and range

See the attached file.

Consider the set of integers Z. The set Z x Z consists of all ordered pairs of integers. In symbols,

Z x Z={(x,y):x,y∈Z}

For example, (2,-5) , (0,0) , (-127,10) , and (-5,2) are all distinct elements of Z x Z.

Notice that the order matters; (2,-5) and (-5,2) are different elements of Z x Z.

Now let?s define a few functions on Z x Z:

f(x,y)=x*y

(See the attached file for full equations)

h(x,y)=x^y

(a) What is the range of each of these functions? In other words, is the image an integer, a rational number or only a real number?

(b) For which of these functions (if any) is the image of (x,y) equal to the image of (y,x)?

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#### Solution Preview

See the attached file.

6. Consider the set of integers. The set x consists of all ordered pairs of integers. In symbols,

x

For example, (2,-5) , (0,0) , (-127,10) , and (-5,2) are all distinct elements of x .

Notice that the order matters; (2,-5) and (-5,2) are different elements of x .

Now let's define a few functions on x :

(a) What is the range of ...

#### Solution Summary

The solution examines Z x Z functions and range.