Math/Physic/Economic/Statistic Problems
HOME WORK TWO
Question One:
The function f:ℝ→ℝ is defined by the rule f(x)=2×2 −3x+1 .
(i) Show that f is not a one-to-one function and that it is not an onto function.
(ii) Which interval of ℝ is the range of the function f ?
(iii) Express the image set f([1,2]) as an interval of ℝ .
(iv) Express the preimage set f−1([0,∞)) as a union of intervals of ℝ .
Question Two:
Let h:A→B be a one-to-one function from a set A to a set B. The function f can be used to
construct a function H:𝒫(A)→𝒫(B) as follows: For each X∈𝒫(A)
H(X)=h(X) (i.e. the image set of 𝑋 , by ℎ ).
Show that 𝐻 is one-to-one ⇔ ℎ is one-to-one.
Question Three:
Let A , B and C be subsets of a universal set. Either prove or provide a counter example for the
following statements:
(i) (A∖(B∩C))∖(B∖C)=A∖B .
(ii) (A∖B)∖(B∖C)=A∖C.
(iii) (A∪B)∖C=A∪(B∖C).
Question Four:
(i) Show that the function f:ℕ×ℕ→ℕ , defined by the rule
f(m.n)=2m3n
is a one-to-one function (injective) function.
(ii) Show that the function g:ℝ∖{3}→ℝ∖{2} , defined by the rule
g(x)= 2x
x−3
is a one-to-one function (injective) function, which is onto. Find the inverse function
g−1 of the function g .
Question Five:
The function f:(0,1)→(0,1) is defined by the rule f(x)= 2x
1+y2 . Show that f is a one-to-one and onto function. Find the rule f−1(x) of the inverse function f−1