When I introduce the concept of function in my college algebra class, I emphasize that the formula is not enough. To really know a function is to know its formula and its shape (graph) and its domain and range and its important points including zeros, y-intercept, minima and maxima, and its behavior when x is very large or small and its vertical asymptotes if any. I was casting about for a problem for a high school math contest and came up with this.
The correct answer would be a horizontal line at with holes at and . I didn’t use the problem but I decided to test Wolfram Alpha which I know has problems along this line.
This query gave a wrong or at least a insufficient answer.
I guess Wolfram Alpha‘s has a different idea about what is important information about a function and its graph. I also tried this query.
Essentially the same incomplete answer.
Finally I tried, as Nick Chura has suggested, asking directly about the domain and got the correct answer.
So Wolfram Alpha has the tools but not the philosophy.
By the way, this is a cool technique for putting holes in function graphs. If you want a hole at say just multiply the function formula by . So has a hole at . I particularly like this function . The vertex point is removed and the function is now differentiable in its entire domain. It is a satisfying exercise to calculate its derivative.