Question
In: Advanced Math
Show that W = {f : R → R : f(1) = 0} together with usual...
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Show that W = {f : R → R : f(1) = 0} together with usual addition and scalar multiplication forms a vector space. Let g : R → R and define T : W → W by T(f) = gf. Show that T is a linear transformation.
Solutions
Colby Messinger answered 2 years agoRelated Solutions
Suppose a function f : R → R is continuous with f(0) = 1. Show that...
Suppose a function f : R → R is continuous with f(0) = 1. Show
that if there is a positive number x0 for which
f(x0) = 0, then there is a smallest positive number p
for which f(p) = 0. (Hint: Consider the set {x | x > 0, f(x) =
0}.)
Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R)....
Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R).
Explain why {f(x) : f(0) = 1} is not.
5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f...
5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f is
injective, but not surjective.
(b) Suppose g : R\{−1} → R\{a} is a function such that g(x) =
x−1, where a ∈ R. Determine x+1
a, show that g is bijective and determine its inverse
function.
Show that the set of all solutions to the differential equation f′′+f=0, where f∈F(R), is a...
Show that the set of all solutions to the differential
equation
f′′+f=0,
where f∈F(R), is a real vector space with respect to the usual
definition of vector addition and scalar multiplication of
functions
# O W L S f(O,W,L,S) 0 0 0 0 0 0 1 0 0 0...
#
O
W
L
S
f(O,W,L,S)
0
0
0
0
0
0
1
0
0
0
1
0
2
0
0
1
0
1
3
0
0
1
1
1
4
0
1
0
0
0
5
0
1
0
1
1
6
0
1
1
0
1
7
0
1
1
1
X
8
1
0
0
0
0
9
1
0
0
1
0
10
1
0
1
0
0
11
1
0
1
1
1
12
1...
R is included in (R-{0} )x(R-{0} ) R = {(x,y) : xy >0} Show that R...
R is included in (R-{0} )x(R-{0} )
R = {(x,y) : xy >0}
Show that R is an equivalent relation and find f its equivalent
classes
Let f : [0, 1] → R and suppose that, for all finite subsets of [0,...
Let f : [0, 1] → R and suppose that, for all finite subsets of
[0, 1], 0 ≤ x1 < x2 < · · · < xn ≤ 1,
we have |f(x1) + f(x2) + · · · + f(xn)| ≤ 1. Let S := {x ∈ [0,
1] : f(x) ̸= 0}. Show that S is countable
Which of the following functions are one-to-one? Group of answer choices f;[−3,3]→[0,3],f(x)=9−x2f;[−3,3]→[0,3],f(x)=9−x2 f;R→R,f(x)=x3f;R→R,f(x)=x3 f;[0,∞]→[0,∞],f(x)=x2f;[0,∞]→[0,∞],f(x)=x2
Which of the following functions are one-to-one?
Group of answer choices
f;[−3,3]→[0,3],f(x)=9−x2f;[−3,3]→[0,3],f(x)=9−x2
f;R→R,f(x)=x3f;R→R,f(x)=x3
f;[0,∞]→[0,∞],f(x)=x2f;[0,∞]→[0,∞],f(x)=x2
A probability density function on R is a function f :R -> R satisfying (i) f(x)≥0...
A probability density function on R is a function f :R -> R
satisfying (i) f(x)≥0 or all x e R and (ii) \int_(-\infty )^(\infty
) f(x)dx = 1. For which value(s) of k e R is the function
f(x)= e^(-x^(2))\root(3)(k^(5)) a probability density function?
Explain.
Let f: [0 1] → R be a function of the class c ^ 2 that...
Let f: [0 1] → R be a function of the class c ^ 2 that
satisfies the differential equation f '' (x) = e^xf(x) for all x in
(0,1). Show that if x0 is in (0,1) then f can not have a positive
local maximum at x0 and can not have a negative local minimum at
x0. If f (0) = f (1) = 0, prove that f = 0
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