Let f : N → N and g : N → N be the functions defined...

Let f : N → N and g : N → N be the functions defined as ∀k ∈ N f(k) = 2k and g(k) = (k/2 if k is even, (k + 1) /2 if k is odd).

(1) Are the functions f and g injective? surjective? bijective? Justify your answers.

(2) Give the expressions of the functions g ◦ f and f ◦ g?

(3) Are the functions g ◦ f and f ◦ g injective? surjective? bijective? Justify your answers.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let f : Z × Z → Z be defined by f(n, m) = n −...

Let f : Z × Z → Z be defined by f(n, m) = n − m a. Is this function one to one? Prove your result. b. Is this function onto Z? Prove your result

Let⇀F and⇀G be vector fields defined on R3 whose component functions have continuous partial derivatives. Furthermore,...

Let⇀F and⇀G be vector fields defined on R3 whose component functions have continuous partial derivatives. Furthermore, assume that ⇀∇×⇀F=⇀∇×⇀G.Show that there is a scalar function f such that ⇀G=⇀F+⇀∇f.

In this question we study the recursively defined functions f, g and h given by the...

In this question we study the recursively defined functions f, g and h given by the following defining equations f(0) = −1 base case 0, f(1) = 0 base case 1, and f(n) = n · f(n − 1) + f(n − 2)^2 recursive case for n ≥ 2. and g(0, m, r, k) = m base case 0, and g(n, m, r, k) = g(n − 1, r,(k + 2)r + m^2 , k + 1) recursive case for...

In this question we study the recursively defined functions f, g and h given by the...

Let f and g be two functions whose ﬁrst and second order derivative functions are continuous,...

Let f and g be two functions whose ﬁrst and second order derivative functions are continuous, all deﬁned on R. What assumptions on f and g guarantee that the composite function f ◦g is concave?

Let f: A→B and g:B→C be maps. (A) If f and g are both one-to-one functions,...

Let f: A→B and g:B→C be maps. (A) If f and g are both one-to-one functions, show that g∘f is one-to-one. (B) If g∘f is onto, show that g is onto. (C) If g∘f is one-to-one, show that f is one-to-one. (D) If g∘f is one-to-one and f is onto, show that g is one-to-one. (E) If g∘f is onto and g is one-to-one, show that f is onto. (Abstract Algebra)

Perform four iterations, if possible, on each of the functions g defined in Exercise 1. Let...

Perform four iterations, if possible, on each of the functions g defined in Exercise 1. Let Po=1 and P(n+1)=g(Pn), for n=0,1,2,3. b. Which function do you think gives the best approximation to the solution? heres the functions g defined in exercise 1. g1(x)=(3+x-2x^2)^1/4 g2(x)=(x+3-x^4/2)^1/2 g3(x)=(x+3/x^2+2)^1/2 g4(x)=3x^4+2x^2+3/4x^3+4x-1

if f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x).

if f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x) (a) Find u'(1) (b) Find v'(5).

3. Let F : X → Y and G: Y → Z be functions. i. If...

3. Let F : X → Y and G: Y → Z be functions. i. If G ◦ F is injective, then F is injective. ii. If G ◦ F is surjective, then G is surjective. iii. If G ◦ F is constant, then F is constant or G is constant. iv. If F is constant or G is constant, then G ◦ F is constant.

Let f : X → Y and g : Y → Z be functions. We can...

Let f : X → Y and g : Y → Z be functions. We can deﬁne the composition of f and g to be the function g◦ f : X → Z for which the image of each x ∈ X is g(f(x)). That is, plug x into f, then plug theresultinto g (justlikecompositioninalgebraandcalculus). (a) If f and g arebothinjective,must g◦ f beinjective? Explain. (b) If f and g arebothsurjective,must g◦ f besurjective? Explain. (c) Suppose g◦ f isinjective....

Subjects