Question

If f:X→Y is a function and A⊆X, then define

f(A) ={y∈Y:f(a) =y for some a∈A}.

(a) If f:R→R is defined by f(x) =x^2, then find f({1,3,5}).

(b) If g:R→R is defined by g(x) = 2x+ 1, then find g(N).

(c) Suppose f:X→Y is a function. Prove that for all B, C⊆X,f(B∩C)⊆f(B)∩f(C). Then DISPROVE that for all B, C⊆X,f(B∩C) =f(B)∩f(C).

Answer #1

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Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i)
Prove that if y > 0, then there is a solution x to the equation
f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove
that the function f : R → R is strictly monotone. (iii) By
(i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why
the derivative of the inverse function,...

Consider the function f:R?R given by f(x,y)=(2-y,2-x).
(b) Draw the triangle with vertices A = (1, 2), B = (3, 1), C =
(3, 2), and the triangle with vertices f(A),f(B),f(C).
(c) Is f a rotation, a translation, or a glide reflection?
Explain your answer.

If f:R→R satisfies f(x+y)=f(x) + f(y) for all x and y and
0∈C(f), then f is continuous everywhere.

A function f”R n × R m → R is bilinear if for all x, y ∈ R n and
all w, z ∈ R m, and all a ∈ R: • f(x + ay, z) = f(x, z) + af(y, z)
• f(x, w + az) = f(x, w) + af(x, z) (a) Prove that if f is
bilinear, then (0.1) lim (h,k)→(0,0) |f(h, k)| |(h, k)| = 0. (b)
Prove that Df(a, b) · (h, k) = f(a,...

Let X=2N={x=(x1,x2,…):xi∈{0,1}} and define
d(x,y)=2∑(i≥1)(3^−i)*|xi−yi|.
Define f:X→[0,1] by f(x)=d(0,x), where 0=(0,0,0,…).
Prove that maps X onto the Cantor set and satisfies
(1/3)*d(x,y)≤|f(x)−f(y)|≤d(x,y) for x,y∈2N.

(9)
(a)Find the double integral of the function f (x, y) = x + 2y
over the region in the plane bounded by the lines x = 0, y = x, and
y = 3 − 2x.
(b)Find the maximum and minimum values of 2x − 6y + 5 subject to
the constraint x^2 + 3(y^2) = 1.
(c)Consider the function f(x,y) = x^2 + xy. Find the directional
derivative of f at the point (−1, 3) in the direction...

Let f : R → R + be defined by the formula f(x) = 10^2−x . Show
that f is injective and surjective, and find the formula for f −1
(x).
Suppose f : A → B and g : B → A. Prove that if f is injective
and f ◦ g = iB, then g = f −1 .

] Consider the function f : R 2 → R defined by f(x, y) = x ln(x
+ 2y). (a) Find the gradient of f(x, y) at the point P(e/3, e/3).
(b) Use the gradient to find the directional derivative of f at
P(e/3, e/3) in the direction of the vector ~u = h−4, 3i. (c) Find a
unit vector (based at P) pointing in the direction in which f
increases most rapidly at P.

For each vector field F~ (x, y) = hP(x, y), Q(x, y)i, find a
function f(x, y) such that F~ (x, y) = ∇f(x, y) = h ∂f ∂x , ∂f ∂y i
by integrating P and Q with respect to the appropriate variables
and combining answers. Then use that potential function to directly
calculate the given line integral (via the Fundamental Theorem of
Line Integrals):
a) F~ 1(x, y) = hx 2 , y2 i Z C F~ 1...

Evaluate the double integral for the function
f(x,
y)
and the given region R.
f(x, y) =
5y + 5x;
R is the rectangle defined by
5 ≤ x ≤ 6
and
2 ≤ y ≤ 4

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