Let f:04 →U14 be defined by f(x) = x². Find the kernel off and show that...
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)?
8. Let F be the group of all functions f : R → R under addition. (a) Let H F be the subgroup of all functions f such that f(0) -0. What group is FH isomorphic to? (Hint: what is H the kernel of?) (b) Let C F be the subgroup of constant functions. Show that F/C is isomorphic to the subgroup H from part (a). (c) Let K F be the subgroup of al functions f that are continuous...
Let T:P R^2 be defined by T(p(x)) = (p(1),p(-1)). (a) Find T(p(x)) where p(x) = 2 + 5x. (b) Show that T is a linear transformation. (C) Find the kernel of T. Explain why T is one-to-one. (d) Find the range of T. Explain why I' is onto. (e) Find T-1(3,7)
Let T(f(t)) = t(f(t)) from P to P. Find the image and kernel of T.
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.
- 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.
Detailed solution please Let f be defined by f(x) = 25(x + 4) For the following, no decimal entries allowed. For parts (d) and (e), remember that you can enter your answer as an expression and let wamap be the calculator. (a) List the critical value(s) of f. If there is more than one, list them separated by commas. (b) Find where f is decreasing. Answer in interval notation. (c) Find where f is increasing. Answer in interval notation. (d)...
7. For fixed x', the Gaussian kernel function f(x ) = - is the solution to Fourier's heat equation f(x|t) = 3072f(x|t), XER, 1 > 0, with initial condition f(x|0) = (x - x') (the Dirac function at x'). Show this. As a con- sequence, the Gaussian KDE is the solution to the same heat equation, but now with initial condition f(x0) = n-' - (x – x;). This was the motivation for the theta KDE [14], which is a...
show work 2. Let f(x) = x* – 18x²+4. a) Find the intervals on which f is increasing or decreasing. b) Find the local maximum and minimum values off. c) Find the intervals of concavity and the inflection points. d) Use the information from a-c to make a rough sketch of the graph.
9. La ste) defined as follows 9. Let f(x) defined as follows: f(x) = 0 if x < -1 = 2(x + 1)/27 if - 1<x<2 = 2/9 if 2 < x < 5 = 0 otherwise. Find F(u) = f(x)dx, where u E R.