Re: Finding set of possible values of a given function

On Feb 21, 3:29 pm, newbar...@xxxxxxxxx wrote:
Ray,

Just a quick note to say thank you. I've been working through your e-
mail for about an hour - especially paragraph 2 about finding L and U.

Think about it graphically. Near the maximum of F(x), the graph y = F
(x) looks like a hilltop. A horizontal line that is too high will miss
the hilltop altogether; this is the case when v is too large, so that
all x give F(x) < v. As we keep lowering v, we get to the scenario
where the line of height v just grazes the hilltop. Now look what
happens if we lower the line just a bit more. The line will now pass
through the hill, entering on one side and leaving on the other. This
would be the case where the equation F(x) = v has two roots (one at
the entry point and the other at the exit point---just like the entry
and exit wounds so commonly mentioned in forensic television dramas).
At the grazing point we want the two roots to coalesce into one. That
would be the case where the quadratic equation has just one root, due
to the vanishing of the +-sqrt() part. The same thing applies to the
minimum, except there we are looking at a valley bottom instead of a
hilltop.

Anyway, the best way to understand what is happening is to make a
simple sketch, drawing a hypothetical graph y = F(x) near the peak
value and looking at horizontal lines y = v for various v near the
maximum.


It's not fully making sense to me,

As I said: draw a picture.

R.G. Vickson


but I think it might be easier
tomorrow as it's late here now.

One thing you do is set the function equal to v. This is really
interesting. I've always seen the function set equal to zero and have
only ever really thought about finding root that are x-intercepts. I
didn't know you could set it equal to another value to find the roots
there. It sounds so fundamental a thing to want to do though.

The next chapter introduces differentiation so at the moment I'm still
pre-calculus. I really want to understand inequalities and quadratics
as much as possible before moving on though.

Will let you know how I get on when I can visit your e-mail again.

Regards,

Pete

.

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