Question

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.

Answer #1

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.

Let f and g be continuous functions on the reals and let S={x in
R | f(x)>=g(x)} . Show that S is a closed set.

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

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)

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...

the values of two functions, f and g, are
given in a table. One, both, or neither of them may be exponential.
Give the exponential models for those that are. HINT [See Example
1.] (If an answer does not exist, enter DNE.)
x
−2
−1
0
1
2
f(x)
0.18
0.9
4.5
22.5
112.5
g(x)
8
4
2
1
0.5
f(x)
=
g(x)
=

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.

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.

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.

Let F be a field and f(x), g(x) ? F[x] both be of degree ? n.
Suppose that there are distinct elements c0, c1, c2, · · · , cn ? F
such that f(ci) = g(ci) for each i. Prove that f(x) = g(x) in
F[x].

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