Question

If R is a UFD and f(x), g(x) contains R[x], prove that c(fg) and c(f)c(g) are associatives.

Answer #1

Prove that if F is a field and K = FG for a finite
group G of automorphisms of F, then there are only finitely many
subfields between F and K.
Please help!

If f, g: R→R are functions such that f(x) = g(x) for all x∈Q and
C(f) = C(g) = R, then f = g.

1. Let f(x)=x2−9 and g(x)=15−x.
Perform the composition or operation indicated.
(fg)(−8)=
2. Let f(x)=x2−8 and g(x)=16−x.
Perform the composition or operation indicated.
(f/g)(7)=
3. Let f(x)=5x+4 and g(x)e=4x−7. Find (f+g)(x),(f−g)(x),
(fg)(x), and (f/g)(x)
Give the domain of each.

let F : R to R be a continuous function
a) prove that the set {x in R:, f(x)>4} is open
b) prove the set {f(x), 1<x<=5} is connected
c) give an example of a function F that {x in r, f(x)>4} is
disconnected

Let f and g be measurable unsigned functions on R^d . Assume
f(x) ≤ g(x) for almost every x. Prove that the integral of f dx ≤
Integral of g dx.

a) Let f : [a, b] −→ R and g : [a, b] −→ R be differentiable.
Then f and g differ by a constant if and only if f ' (x) = g ' (x)
for all x ∈ [a, b].
b) For c > 0, prove that the following equation does not have
two solutions. x3− 3x + c = 0, 0 < x < 1
c) Let f : [a, b] → R be a differentiable function...

Let f: R -> R and g: R -> R be differentiable, with g(x) ≠
0 for all x. Assume that g(x) f'(x) = f(x) g'(x) for all x. Show
that there is a real number c such that f(x) = cg(x) for all x.
(Hint: Look at f/g.)
Let g: [0, ∞) -> R, with g(x) = x2 for all x ≥ 0. Let L be
the line tangent to the graph of g that passes through the point...

Let A, B ⊆R be intervals. Let f: A →R and g: B →R be
diﬀerentiable and such that f(A) ⊆ B. Recall that, by the Chain
Rule, the composition g◦f: A →R is diﬀerentiable as well, and the
formula
(g◦f)'(x) = g'(f(x))f'(x)
holds for all x ∈ A. Assume now that both f and g are twice
diﬀerentiable.
(a) Prove that the composition g ◦ f is twice diﬀerentiable as
well, and ﬁnd a formula for the second derivative...

Prove that f : R → R where f(x) = |x| is neither injective nor
surjective.

Prove that the function f : R \ {−1} → R defined by f(x) = (1−x)
/(1+x) is uniformly continuous on (0, ∞) but not uniformly
continuous on (−1, 1).

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