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 tha...
- 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.
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]
Let A be n × n with AT-A. (The matrix A is syrnmetric.) Let B be 1 × n and let c E R. Define f : Rn → R by f(x) = 2.7, A . x + B . x + c. Show that The function f is a quadratic function
Let A be n × n with AT-A. (The matrix A is syrnmetric.) Let B be 1 × n and let c E R. Define f : Rn...
set theory:
Let f: R - R be defined by f(x) = x2 – 40. Find the following: a) the image of -1, b) the pre-image of -3.
2. Consider f(x)={ x2 sin (1) xメ0 x) = (a) Show the function has a derivative for xE [0,1 (b) Show the function does not have a second derivative for x E [0,1] (c) Does this violate our understanding of holomorphic functions?
Problem 5. Given vi,v2,... ,Vm R", let RRm be defined by f(x)-x, v1), x, v2), (x, Vm where (x' y) is the standard inner product of Rn Which of the following statement is incorrect? 1. Taking the standard bases Un on R": codomain: MatUn→Un(f)-(v1 2. Taking the standard bases Un on R: codomain: v2 vm) Matf)- 3. f is a linear transformation. 4. Kerf- x E Rn : Vx = 0 , where: Problem 8. Which of the following statements...
Let f : [0, 1] x [0, 1] + R be defined by f(x, y) = {1 if y = 23, 0 if y + x2 Show that f is integrable on (0, 1] x [0, 1]. You may take the previous problem as given
Let f : [0, 1] x [0, 1] → R be defined by f(x,y) - 1 if y=%, 0 if y#x2 Show that f is integrable on [0,1] [0,1]. You may take the previous problem as given
Exercise 17.10 Let x = V2 and for n > 1 let In+1 = 2 + In Use Banach's fixed point theorem to show that (en) converges to a root of the equation r' - 4x2 - + 4 = 0 lying between 3 and 2.
Let x = [xı x2 x3], and let TER → R be the linear transformation defined by T() = x1 + 6x2 – x3 -X2 X1 + 4x3 Let B be the standard basis for R2 and let B' = {V1, V2, V3}, where 7 7 and v3 = 7 V1 V2 [] --[] 0 Find the matrix of I with respect to the basis B. and then use Theorem 8.5.2 to compute the matrix of T with respect to...