Question

Determine whether each of the following functions is an injection, a surjection, both, or neither:

(a) f(n) = n^3 , where f : Z → Z

(b) f(n) = n − 1, where f : Z → Z

(c) f(n) = n^2 + 1, where f : Z → Z

Answer #1

3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine whether f is onto. Prove your
answers.
(a) f : R → R by f(x) = x^2 if x ≥ 0, 2x if x < 0.
(b) f : Z → Z by f(n) = n + 1 if n is even, 2n if n is odd.

Let A be a finite set and let f be a surjection from A to
itself. Show that f is an injection.
Use Theorem 1, 2 and corollary 1.
Theorem 1 : Let B be a finite set and let f be a function on B.
Then f has a right inverse. In other words, there is a function g:
A->B, where A=f[B], such that for each x in A, we have f(g(x)) =
x.
Theorem 2: A right inverse...

For each of the following pairs of functions f and g (both of
which map the naturals N to the reals R), state whether f is O(g),
Ω(g), Θ(g) or “none of the above.” Prove your answer is correct. 1.
f(x) = 2 √ log n and g(x) = √ n. 2. f(x) = cos(x) and g(x) =
tan(x), where x is in degrees. 3. f(x) = log(x!) and g(x) = x log
x.

For each of the following pairs of functions f and g (both of
which map the naturals
N to the reals R), show that f is neither O(g) nor Ω(g). Prove
your answer is correct.
1. f(x) = cos(x) and g(x) = tan(x), where x is in
degrees.

Determine which of the following functions are injective,
surjective, bijective (bijectivejust means both injective and
surjective).
(a)f:Z−→Z, f(n) =n2.
(d)f:R−→R, f(x) = 3x+ 1.
(e)f:Z−→Z, f(x) = 3x+ 1.
(g)f:Z−→Zdefined byf(x) = x^2 if x is even and (x −1)/2 if x is
odd.

determine whether each of the following functions are one-to-one
by using the horizontal line test.
(a) f(x) = x2 + 5
Yes, it is one-to-one. No, it is not
one-to-one.
(b) g(x) = 3x3 + 2
Yes, it is one-to-one.No, it is not
one-to-one.
(c) h(x) = |x - 2|
Yes, it is one-to-one.No, it is not
one-to-one.

Determine whether the following production functions have
constant returns to scale, decreasing returns to scale, or
increasing returns to scale: a) f(x1, x2)= 2x1 0.6x2 0.6
b) f(x1, x2)= x1+min(x1, x2)
c) f(x1, x2)= x1 1/2+x2 1/2

Determine whether the given equation is separable, linear,
neither, or both.
3r=dr/dx-5x^3

For each of the following production functions, (i) sketch an
isoquant, (ii) indicate whether each marginal product is
diminishing, constant, or increasing points, and (iii) indicate
whether the production function shows constant, decreasing, or
increasing returns to scale.
A) Q = f(L, K) = L2K3
B) Q = f(L, K) = 2L +
6K
C) Q = f(L, K) = (minL, K) (1/2)

1. Determine whether each of these production functions has
constant, decreasing, or increasing returns to scale:
(a) F(K,L)= K^2/L
(b) F(K,L)=K+L
2. Which of these production functions have diminishing marginal
returns to labor?
a) F(K,L)=2K+15L
b) F(K,L)=√KL
c) F(K,L)=2√K+15√L
3. For any variable X,
ΔX = “change in
X ”, Δ is the Greek (uppercase) letter
delta, Examples: If ΔL = 1 and
ΔK = 0, then ΔY
= MPL.
More generally, if ΔK = 0, then
Δ(Y − T...

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