Re: Finding set of possible values of a given function

Ray,

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.

That all makes sense and think I already understood most of it.

I've gone over your paragraph 2 from your previous e-mail again and
that's sunk in well now. A good night's sleep does help!

There's something still not clicking into place though. Maybe I can
give another example which doesn't intercept the x-axis to show what I
mean:

y = (1 + x^2) / x.

Changing it into a quadratic in 'x':

x^2 - yx + 1 = 0.

I know that values that satisfy the discriminant >=0 identify the
range of the output.

b^2 - 4ac >=0
--> y^2 - 4 >=0
--> roots are -2 and 2 and the inequality is satisfied when y <= -2
and y >= 2.

But I don't know why. Previously the book's instructed me to inspect
the discriminant of a quadratic and > 0 means 2 x-intercepts, = 0
means one x-intercept (like y = x^2 just touches the x-axis), and < 0
means a non-real/complex x-intercepts (which I've not really learnt
about yet).

So how does applying discriminant >= 0 work when looking at possible
values of y = (1 + x^2) / x?

Thanks for your help so far!

Regards,

Pete
.

Relevant Pages

  • Re: Finding set of possible values of a given function
    ... looks like a hilltop. ... and exit wounds so commonly mentioned in forensic television dramas). ... At the grazing point we want the two roots to coalesce into one. ... would be the case where the quadratic equation has just one root, ...
    (sci.math)
  • Re: Finding set of possible values of a given function
    ... looks like a hilltop. ... At the grazing point we want the two roots to coalesce into one. ... drawing a hypothetical graph y = Fnear the peak ... Draw a picture. ...
    (sci.math)
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