Question

Prove: If D = Q\{3}. R = Q\{-3}, and f:D-> R is defined by f(x) = 1+3x/3-x for all x in D, then f is one-to-one and onto.

Answer #1

Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i)
Prove that if y > 0, then there is a solution x to the equation
f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove
that the function f : R → R is strictly monotone. (iii) By
(i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why
the derivative of the inverse function,...

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).

Let f : R − {−1} →R be defined by f(x)=2x/(x+1).
(a)Prove that f is injective.
(b)Show that f is not surjective.

Which of the following are one-to-one, onto, or both?
a. f : Q → Q defined by f(x) = x3 + x.
b. f : S → S defined by f(x) = 5x + 3.
c. f : S → S defined by:
?(?) = {
? + 1 ?? ? ≥ 0
? − 1 ?? ? < 0 ??? ? ≠ −10
? ?? ? = −10
d. f : N → N × N defined by f(n)...

Suppose f: R^2--->R is defined by f(x,y) = 3y. Is f
one-to-one? Is f onto? Is f a bijection?

let
T: P3(R) goes to P3(R) be defined by T(f(x))= xf'' (x) + f'(x).
Show that T is a linear transformation and determine whther T is
one to one and onto.

Is the function f : R → R defined by f(x) = x 3 − x injective,
surjective, bijective or none of these?
Thank you!

Let f : R → R + be defined by the formula f(x) = 10^2−x . Show
that f is injective and surjective, and find the formula for f −1
(x).
Suppose f : A → B and g : B → A. Prove that if f is injective
and f ◦ g = iB, then g = f −1 .

Consider the function f defined on R by f(x) = ?0 if x ≤ 0, f(x)
= e^(−1/x^2) if x > 0.
Prove that f is indefinitely differentiable on R, and that
f(n)(0) = 0 for all n ≥ 1. Conclude that f does not have a
converging power series expansion En=0 to ∞[an*x^n] for x near the
origin. [Note: This problem illustrates an enormous difference
between the notions of real-differentiability and
complex-differentiability.]

3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine whether f is onto. Prove your
answers.
(a) f : R → R by f(x) = x^2 if x ≥ 0, 2x if x < 0.
(b) f : Z → Z by f(n) = n + 1 if n is even, 2n if n is odd.

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