- #1

- 1,344

- 32

## 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?