Graph Sketching and Proving Restrictions on a Real Function's Range

[spaghetti3451]

Homework Statement

Sketch the graph of the function y(x) = (x-3)/ [(x+1)*(x-2)], indicating the positions of the turning points. Prove that there is a range of values which y can't take if x is real.

Homework Equations

The Attempt at a Solution

To draw the graph, I found

1. the vertical asymptotes which are x = -1 and x = 2.
2. As x tends to -1 from the left, y tends to -ve infinity.
As x tends to -1 from the right, y tends to +ve infinity.
As x tends to 2 from the left, y tends to +ve infinity.
As x tends to 2 from the right, y tends to -ve infinity.
3. As x tends to -ve infinity, y tends to 0 from below the x-axis.
As x tends to +ve infinity, y tends to 0 from above the x-axis.
4. The turning points are (1,1) and (5,1/9).

The graph can be drawn using 1-4.

I think so far I have got everything right. The problem is with proving that there is a range of values which y can't take if x is real.

I considered the x-axis number line in chunks:

1. x < -1 : y < 0.
2. -1 < x < 2 : y > 1.
3. x > 2 : y < 1/9.

This shows that 1/9 < y < 1 is not in the range if the domain consists of real x.

Does this constitute a valid proof?

 

[scurty]

I agree with all of the work you have done. I would just add a little more substance to your proof, why is y great than or less than those numbers? I would say something about the critical points (or turning points as you call them).