Expert: Scotto - 2/27/2009

Question
Scotto can you please help me on this question:

Consider this function y=3x-1/2x on restricted domain D={x|x<0}. Find the range of this function on the restricted domain.

Answer
To find the range, not that the domain is all x in (-infinity,0).
I will assume the function is y = 3x - 1/(2x).

If you take the -infinity and put it into the equation,
it can be noted that the second term is 0,
so the answer is - infinity.

If you take 0 and put it in for x, you get -infinity.

If we let y=f(x) and find df/dx, we get 3 + 1/(2x²).  
This is always positive.

Note that at 0, the function is undefined.
There is a vertical assymptote at 0 and the function is going up.

It looks like the range of this function is all numbers.

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Now if what was meant was (3x-1)/(2x), that is a totally different case.  Since x is negative, as x goes to 0, you get a negative
over a negative, which is a positve.  You also tend to -1/0, but
note that the bottom is always negative, so you get infinity.

As x goes to -infinity, all we need to do is look at the
exponets on the x values, so the function approaches 3/2.

Taking the derivative is a little bit more complicated here.
Remember the quotient rule?  It's (lo d hi - hi d lo)/lo².
That would be (2x*3 - (3x-1)2)/(4x²).

You can see in the numerator, you get 6x and a -6x, so they cancel.
This only leaves you with a constant, so the derivative is never 0.
This means there are no maximums or minimums anywhere in the
function.

What we have, then, is a curve that slopes from a level 3/2 down to
the bottom of the y axis.