Question

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.

Answer #1

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 .

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

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

8.4: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is surjective then g is surjective.
8.5: Let f : X → Y and g : Y→ Z be bijections. Prove that if
composition g o f is bijective then f is bijective.
8.6: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is bijective then f is...

Let X, Y and Z be sets. Let f : X → Y and g : Y → Z functions.
(a) (3 Pts.) Show that if g ◦ f is an injective function, then f is
an injective function. (b) (2 Pts.) Find examples of sets X, Y and
Z and functions f : X → Y and g : Y → Z such that g ◦ f is
injective but g is not injective. (c) (3 Pts.) Show that...

Let f:A→B and g:B→C be maps. Prove that if g◦f is a bijection,
then f is injective and g is surjective.*You may not use, without
proof, the result that if g◦f is surjective then g is surjective,
and if g◦f is injective then f is injective. In fact, doing so
would result in circular logic.

Prove that: If f : R → R is strictly increasing, then f is
injective.

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 → 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,...

Let f : A → B and g : B → C. For each statement below either
prove it or construct f, g, A, B, C which show that the statement
is false.
(a) If g ◦ f is surjective, then g is surjective.
(b) If g ◦ f is surjective, then f is surjective.
(c) If g ◦ f is injective, then f and g are injective

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