Question

Determine which of the following functions are injective (one-to-one) on their respective domains and codomains

(a) f : ℝ → [0,∞), where f(x) = x²

(b) g : ℕ → ℕ, where g(x) = 3x − 2

(c) h : ℤ_7 → ℤ_7, where h(x) ≡ 5x + 2 (mod 7)

(d) p : ℕ ⋃ {0} → ℕ ⋃ {0}, where p(x) = x div 3

Answer #1

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 the intervals on which the following functions are
increasing and decreasing.
a) f(x) = 12 + x - x^2
b) g(x) = 2x^5 - (15x^4/4) + (5x^3/3)

2) Characterize the following functions in terms of whether they
are injective, surjective, and/or bijective. If possible, find
their inverses.
a. ?:ℝ+ → ℝ+ with ?(?) = ?3
b. ?:ℤ+ → ℤ+ with ?(?) = ?3
c. ?:? → ? with ? being the set of calendar dates and ?(?) being
the day numbered ? in the next month that has that day. For
instance, if ? is January 22, 1981 and ? is March 31, 1993, ?(?) is
February...

1. Differentiate the following functions. Do not simplify.
(a) f(x) = x^7 tan(x)
(b) g(x) = sin(x) / 5x + ex
(c) h(x) = (x^4 + 3x^2 - 6)^5
(d) i(x) = 4e^sin(9x)
(e) j(x) = ln(x) / x5
(f) k(x) = ln(cot(x))
(g) L(x) = 4 csc^-1 (x2)
(h) m(x) = sin(x) / cosh(x)
(i) n(x) = 2 tanh^-1 (x4 + 1)

Suppose the following functions have inverses on their
domains.
(1) f(x) = 4ecos(4x). Find and explicit
formula for f-1(y).
(2) g(x) = log3(tan(x)). Find an explicit
formula for g-1(y).

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.

3) If A = 3
1 and B
= 1 7
0
-2
5 -1
Find
a) BA
b) determinant
B
c) Adjoint A
d)
A-1
4) Using matrix method solve the following simultaneous
equations
5x – 3y = 1
2x – 2y = -2
5) Given that f(x) = 6x - 5 g(x) = 3x +
4 and h(x) = 4x – 6
2
Find:-
i)...

Determine whether the given set ?S is a subspace of the vector
space ?V.
A. ?=?2V=P2, and ?S is the subset of ?2P2
consisting of all polynomials of the form
?(?)=?2+?.p(x)=x2+c.
B. ?=?5(?)V=C5(I), and ?S is the subset of ?V
consisting of those functions satisfying the differential equation
?(5)=0.y(5)=0.
C. ?V is the vector space of all real-valued
functions defined on the interval [?,?][a,b], and ?S is the subset
of ?V consisting of those functions satisfying
?(?)=?(?).f(a)=f(b).
D. ?=?3(?)V=C3(I), and...

1. Create a script named AnonIntegrals.m. Within it, define each
of the following functions as anonymous functions and use the
integral command to compute its definite integral over the domain
given. Display the integral calculation to the command window. (a)
p(x) = 4x 2 − 1, x ∈ [0, 1] (b) q(x) = sin(x), x ∈ [0, π] (c) r(x)
= cos(x), x ∈ [−π/2, π/2] (d) s(x) = log(x), x ∈ [0, 1] (e) t(x) =
1 x ,...

Question 6) Suppose X is a random variable taking on possible
values 0,2,4 with respective probabilities .5, .3, and .2. Y is a
random variable independent from X taking on possible values 1,3,5
with respective probabilities .2, .2, and .6. Use R to determine
the following.
f) Find the expected value of X*Y. (i.e. X times Y)
g) Find the expected value of 3X - 5Y.
h) Find the variance of 3X - 5Y
i) Find the expected value of...

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