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

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

Wolfram Alpha Example 2

Essentially the same incomplete answer.

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

Wolfram Alpha Example 3

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.


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.
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