How to find range of a rational function without using derivatives

Question

I am a TA for a pre-calc class. They only know how to draw graphs of lines, parabolas, etc. and don't know differentiation. How can I find the range of a rational funcion like $f(x)= \frac{(x+5)}{(x-1)(x+2)}$ without using differentiation? Also how can we sketch its graph again w/o using first or second derivative?

Answer 1

The given function $f(x)$ satisfies

$$\lim_{x\rightarrow \pm\infty} f(x)=0$$

It has a root and the sign changes at both poles. Without further information, you can only conclude that the range contains all positive numbers and all negative numbers less than or equal some negative number $y$.

You need differentation to find the local extrema, otherwise, I do not think that you can find out the range.

A sketch of the graph is possible without differentiation, but it is very roughly. What can be concluded here is that the function must have a local maximum in the interval $[-2,1]$ and a local minimum in $(-\infty,-5]$

Answer 2

With rational functions, you don't really need calculus to get a good graph. The asymptotes are a nice skeleton on which to hang the graph. In your example, the vertical asymptotes divide the plane into 3 vertical regions. By plotting only one test point in each region, and thinking about where the function is positive and negative, one can tell how the function lies in the asymptotes.

You have an $x$-intercept at $-5$, and a vertical asymptote at $x=-2$. The function is negative for $x< -5$ and positive for $-5<x<-2$, so the graph comes in from 0 at $-\infty$ decreases to some point, then turns and hits the $x$-axis at $-5$ and then proceed to $+\infty$ in the $y$ direction as $x$ nears $-2$.

Similarly, between $-2<x<1$, the function is negative and the $y$-intercept is $-5/2$, the graph in this region is an infinite hotdog stretching down to $-\infty$. For $x>1$, the function is positive and must be asymptotic to $0$ as $x\to\infty$. So it must look like the positive half of $1/x$.

You said in a comment that you'd prefer to do the ranges algebraically. I don't know a good way to do this. I always graph and then just look to see what the range is. In your example, there is a local min to the left of $x=-5$, but it's above the local max between $-2$ and $1$. So there's a gap in the range, but I don't think you can find it without calculus or a whopping lot of lucky guessing. I'd have the students graph and eye-ball it. And I wouldn't even point out the error if they thought the local max was at $x=0$.