
- Let f be the function from R to R defined by f(x)=x2.Find a) f−1({1}). b) f−1({x | 0 < x < 1} c) f−1({x|x>c) f−1({x|x>4}). -Show that the function f (x) = e x from the set of real numbers to the set of real numbers is not invertible but if the codomain is restricted to the set of positive real numbers, the resulting function is invertible.
Set theory
Find f(A) and f-(B) for the given function and sets. (a) f : R → R is defined by f(x) = x2 + 1, A = [-1,2], B = [0,4]. (b) f R R is defined by f(x) - sin z, A [0, ], B [0,2]. (c) f : R → Z is defined by f(x)-번 (the floor function), A = (0,4], B-(0,1,2). (d) f : R-(0) → R is defined by f(z) = x + 1, As...
I. Let f : R → R be defined by f(x)-x2 +1. Determine the following (with minimal explanation): (a) f(I-1,2]) 1(I-1,2 (c) f(f3,4,5) (d) f1(3,4,5)) (e) Is 3 € f(Q)? (f) Is 3 є f-1 (Q)? (g) Does the function f1 exist? If so describe it (h) Find three sets, A R such that f(A)-[5, 17]
3. Let f(r) be defined by and let F(x) be defined by F(x) = Í f() dt, a. Find F(x). 0 x 2. For what value of b in the definition of f is F(x) differentiable for all x E [0, 2)?
Question 3 Let the function f be defined by f(x,y)--3y3 +4y2-15y +x2-8x. The set A consists of all points (x,y) in the xy-plane that satisfy 0sx S 10, 0sy s10 and x +y28.Find the global minimum value of f(x,y) over the set A. (Hint: see example 8 in lecture 7.) (6 marks)
Question 3 Let the function f be defined by f(x,y)--3y3 +4y2-15y +x2-8x. The set A consists of all points (x,y) in the xy-plane that satisfy 0sx S 10,...
Let f:R2→R be defined by f(x,y) =|xy|e−(x2+y2). Evaluate
∫R2f, if it exists
Let f : R2 + R be defined by f(L,y) = [tyle=(3++y?). Evaluate Sir2 f, if it exists
Let the function f be defined by f(x,y)-- уз +4y2-15y + x2-8x . The set A consists of all points (x,y) in the xy-plane that satisfy 0sx s 10, 0sy s10 and x+y 28. Find the global minimum value of f(x,y) over the set A. (Hint: see
Let the function f be defined by f(x,y)-- уз +4y2-15y + x2-8x . The set A consists of all points (x,y) in the xy-plane that satisfy 0sx s 10, 0sy s10 and x+y...
Let F, C R be defined by F.-{x | x 20 and 2-1/n-x2〈 2+1/n). Show that n-&メ2. Use this to show the existence of V2. 18.
Let F, C R be defined by F.-{x | x 20 and 2-1/n-x2〈 2+1/n). Show that n-&メ2. Use this to show the existence of V2. 18.
#10. Define f: R ® R defined by f(x) = 4x2. Let A = [0, 4] and B = [-3, 0]. #10(a) Find the sets f(A), f(B), f(A) Ç f(B), and f(A Ç B). Is f(A Ç B) = f(A) Ç f(B)? #10(b) Find the sets which are inverse images: f -1(A), f -1 (B), f -1 (A) Ç f -1 (B), and f -1 (A Ç B). Is f -1 (A Ç B) = f -1 (A) Ç f...
Let h : X −→ Y be defined by
h(x) :=
f(x) if x ∈ F
g
−1
(x) if x ∈ X − F
Now we must prove that h is injective and bijective. Starting
with injectivity, let x1, x2 ∈
X such that h(x1) = h(x2). Assume x1 ∈ F and x2 ∈ X −F. Then h(x1)
= f(x1) ∈ f(F)
and h(x2) = g
−1
(x2) ∈ g
−1
(X − F) = Y...