Question

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.

Answer #1

1. A function f : Z → Z is defined by f(n) = 3n − 9.
(a) Determine f(C), where C is the set of odd integers.
(b) Determine f^−1 (D), where D = {6k : k ∈ Z}.
2. Two functions f : Z → Z and g : Z → Z are defined by f(n) =
2n^ 2+1 and g(n) = 1 − 2n. Find a formula for the function f ◦
g.
3. A function f :...

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,...

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

Let f, g : Z → Z be defined as follows:
? f(x) = {x + 1 if x is odd; x - 1 if x is even}, g(x) = {x - 1
if x is odd; x + 1 if x is even}.
Describe the functions fg and gf. Then compute the orders of f,
g, fg, and gf.

Which of the following are one-to-one, onto, or both?
a. f : Q → Q defined by f(x) = x3 + x.
b. f : S → S defined by f(x) = 5x + 3.
c. f : S → S defined by:
?(?) = {
? + 1 ?? ? ≥ 0
? − 1 ?? ? < 0 ??? ? ≠ −10
? ?? ? = −10
d. f : N → N × N defined by f(n)...

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.

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.

Prove: If D = Q\{3}. R = Q\{-3}, and f:D-> R is defined by
f(x) = 1+3x/3-x for all x in D, then f is one-to-one and onto.

Consider the piecewise defined function
f(x) = xa− xb if 0<x<1. and
f(x) = lnxc if x≥1.
where a, b, c are positive numbers chosen in such a way that
f(x) is differentiable for all 0<x<∞. What can be said about
a, b, and c?

For each of the following functions fi(x), (i) verify that they
are legitimate probability density functions (pdfs), and (ii) find
the corresponding cumulative distribution functions (cdfs) Fi(t),
for all t ? R.
f1(x) = |x|, ? 1 ? x ? 1
f2(x) = 4xe ?2x , x > 0
f3(x) = 3e?3x , x > 0
f4(x) = 1 2? ? 4 ? x 2, ? 2 ? x ? 2.

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