Confused Domain(?) Real Or Imaginary

A Complex Number Set And Function Story

I asked my students to graph     y = x² + 1     and they responded with the graph:

y = x^2 + 1


I asked my students to find the zeros of

      y = x² + 1

and they responded:

      x² + 1 = 0       is       x² - i² = 0

      (x + i)(x - i) = 0       so

      x = -i     or     x = i

since we talked about complex number arithmetic before this question.

In this solution   x   is not a real number!

In fact, they made   x   a complex number   w = a + b i   with   a = 0 ,   b = 1   and   i² = -1.

When my students used complex number arithmetic to solve this question, I was pleased that they had learned their complex number material well.     Good for them!!

But they extended the domain of   y = x² + 1   from all real numbers to all real numbers and two complex numbers. We notice that the union of this set of two imaginary numbers with the real number domain does not change the range by including complex numbers with non zero imaginary parts. The range is the union of the set containing zero with all real numbers greater than or equal to   1. The range is a real number set.

I looked forward to the next class and finding students who did or would wonder "Why not have a   y   output that could be a complex number for different complex number   x   selections?"

What can happen to the domain of   y = x² + 1   and how does that change the range of   y = x² + 1 ?

Let   z   and   w   be complex number variables for the function   z = w² + 1   and   x   and   y   be real number variables for the function   y = x² + 1   then we just covered some of   y = w² + 1   above.

Consider different number set possibilities: (click on the box)

                         DOMAIN
RANGE
REAL NUMBERS   x   COMPLEX NUMBERS   w  
REAL NUMBERS   y   A. NO ZEROS B. SOME ZEROS
COMPLEX NUMBERS   z   C. NO POINTS D. WHY NOT?

Confused Domain Answer

If you let   x   and   y   be real number variables for the function   y = x² + 1   then the above graph helps us see the that:
  •     The domain for   y   is the set of all   x   such that   x   is any real number.
  •     The range is the set of all   y   such that   y   is any real number greater than or equal to   1.
  • When asked the question:

            "What are the zeros of   y = x² + 1?"

    you can say they are:   x = -i   or   x = i   if you let   x   be a complex number.

    Then consider if these two zeros are to be included in the domain of   y.

    If the result of this consideration is yes, then you may wish to click on B in the table above.



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