A Tricky Domain Problem and Wolfram Alpha

What is the domain of the function f(x)=\frac{1}{\frac{x-3}{x-4}}?  The answer is \{x|x \neq 3 \: and \: x \neq 4\} because if x = 3 or x= 4 a divide by zero situation occurs.  One is sorely tempted to simplify f(x) to f(x)=\frac{x-4}{x-3} which would be incorrect as it stands since now x\neq 4 needs to be explicitly stated.  So f(x)=\frac{x-4}{x-3},x\neq 4 .

Wolfram Alpha handles this function curiously.  If the original functions is input as f(x)=\frac{1}{\frac{x-3}{x-4}},  Alpha simplifies it to  f(x)=\frac{x-4}{x-3} with nary a mention that x\neq 4.  This is clearly wrong.  Alpha also shows the graph of the function as if f(4) exists instead of displaying a hole.

If the expression \frac{1}{\frac{x-3}{x-4}} is entered, Alpha still simplifies it to \frac{x-4}{x-3} and shows a function plot without a hole at (4,0). However in a box entitled “Properties as a real function” the domain \{x\epsilon\Re|x \neq 3 \: and \: x \neq 4\} is correct.  I have alluded to problems with Wolfram Alpha in this post and this post.


About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
This entry was posted in Curriculum, Math Explorations. Bookmark the permalink.

One Response to A Tricky Domain Problem and Wolfram Alpha

  1. Nick says:

    Different yet, you could as Wolfram|Alpha for the domain of the function:


    This time it answers correctly with a line graph of the domain.

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