## What to Know About a Function

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.

Graph $f(x)=\frac{(x-3)}{(x+2)}\frac{(x+2)}{(x-3)}$

The correct answer would be a horizontal line at $y=1$ with holes at $x=-2$ and $x=3$.  I didn’t use the problem but I decided to test Wolfram Alpha which I know has problems along this line.

Wolfram Alpha Example 1

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.

Wolfram Alpha Example 2

Finally I tried, as Nick Chura has suggested, asking directly about the domain and got the correct answer.

Wolfram Alpha Example 3

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 $x=2$ just multiply the function formula by $\frac{(x-2)}{(x-2}$.  So $|x|\frac{(x-2)}{(x-2}$ has a hole at $(2,2)$.  I particularly like this function $f(x) =|x|\frac{(x)}{(x}$.  The vertex point is removed and the function is now differentiable in its entire domain.  It is a satisfying exercise to calculate its derivative.