Question

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.

Answer #1

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.

1. a True or False? If ∫ [ f ( x ) ⋅ g ( x ) ] d x = [ ∫ f ( x )
d x ] ⋅ [ ∫ g ( x ) d x ]. Justify your answer.
B. Find ∫ 0 π 4 sec 2 θ tan 2 θ + 1 d θ
C. Show that ∫ 0 π 2 sin 2 x d x = ∫ 0 π 2 cos...

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.

Find each of the following functions. f(x) = 4 − 4x, g(x) =
cos(x)
(a) f ∘ g and State the domain of the function. (Enter your
answer using interval notation.)
(b) g ∘ f and State the domain of the function. (Enter your
answer using interval notation.)
(c) f ∘ f and State the domain of the function. (Enter your
answer using interval notation.)
(d) g ∘ g and State the domain of the function. (Enter your
answer using...

For each of the following pairs of polynomials f(x) and g(x),
write f(x) in the form
f(x) = k(x)g(x) + r(x)
with deg(r(x)) < deg(g(x)).
a) f(x) = x^4 + x^3 + x^2 + x + 1 and g(x) = x^2 −
2x + 1.
b) f(x) = x^3 + x^2 + 1 and g(x) = x^2 − 5x + 6.
c) f(x) = x^22 − 1 and g(x) = x^5 − 1.

1a.
Find the domain and range of the function. (Enter your answer
using interval notation.)
f(x) = −|x + 8|
domain=
range=
1b.
Consider the following function. Find the composite
functions
f ∘ g
and
g ∘ f.
Find the domain of each composite function. (Enter your domains
using interval notation.)
f(x) =
x − 3
g(x) = x2
(f ∘ g)(x)=
domain =
(g ∘ f)(x) =
domain
are the two functions equal?
y
n
1c.
Convert the radian...

For f(x) = x^2+6 and g(x) = x^2-5 find the following
functions.
a.) (f o g)(x)
b.) (g o f) (x)
c.) (f o g) (4)
d.) (g o f) (4)

Find the derivatives of each of the following functions. DO NOT
simplify your answers.
(a) f(x) = 103x (3x5+ x − 1)4
(b) g(x) = ln(x3 + x) /
x2 − 4
(c) h(x) = tan-1(xex)
(d) k(x) = sin(x)cos(x)

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

For functions f and g, where f(x) = 1 + x 2 , g(x) = x − 1 in
C[0, 1] with the inner product defined by the integral: hf , gi = Z
1 0 f(x)g(x)dx, (a) find the norm of f and g. (b) find unit vectors
in the directions of f and g. (c) find the cosine of the angle θ
between f and g. (d) find the orthogonal projection of f along
g

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