Prove that if f is a bounded function on a bounded interval
[a,b] and f is continuous except at finitely many points in [a,b],
then f is integrable on [a,b]. Hint: Use interval additivity, and
an induction argument on the number of discontinuities.
Prove the following statements!
1. If A and B are sets then
(a) |A ∪ B| = |A| + |B| − |A ∩ B| and
(b) |A × B| = |A||B|.
2. If the function f : A→B is
(a) injective then |A| ≤ |B|.
(b) surjective then |A| ≥ |B|.
3. For each part below, there is a function f : R→R that is
(a) injective and surjective.
(b) injective but not surjective.
(c) surjective but not injective.
(d)...
The function f(x)=3x+2 is one-to-one
a) find the inverse of f
b) State the domain and range of f
c) State the domain and range of f-1
d) Graph f,f-1, and y=x on the same set of axes
Consider the function. f(x) = x^2 − 1, x ≥ 1
(a) Find the inverse function of f.
f ^−1(x) =
(b) Graph f and f ^−1 on the same set of coordinate axes.
(c) Describe the relationship between the graphs. The graphs of
f and f^−1 are reflections of each other across the line ____answer
here___________.
(d) State the domain and range of f and f^−1. (Enter your
answers using interval notation.)
Domain of f
Range of f
Domain...
"•“"Suppose ℱ, ?1, and ?2 are nonempty families of sets. Prove
that if ℱ ⊆ ?1 ∩ ?2, then ∩?1 ∪ ∩?2 ⊆ ∩ℱ.“"
Please explain what the question is asking for and break down
the solution for me step by step with explanations please!
Prove
1. Let f : A→ B and g : B → C . If g 。 f is one-to-one, then f
is one-to-one.
2. Equivalence of sets is an equivalence relation (you may use
other theorems without stating them for this one).