Math/Physic/Economic/Statistic Problems

 
Question One:

The function f: is defined by the rule f(x)=2x2 3x+1 .

(i) Show that f is not a onetoone function and that it is not an onto function.

(ii) Which interval of is the range of the function f ?

(iii) Express the image set f([1,2]) as an interval of .

(iv) Express the preimage set f1([0,)) as a union of intervals of .


Question Two:

Let h:AB be a onetoone 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 X𝒫(A)

H(X)=h(X) (i.e. the image set of 𝑋 , by ).

Show that 𝐻 is onetoone is onetoone.

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(BC))(BC)=AB .

(ii) (AB)(BC)=AC.

(iii) (AB)C=A(BC).


Question Four:

(i) Show that the function f:× , defined by the rule

f(m.n)=2m3n

is a onetoone function (injective) function.

(ii) Show that the function g:{3}{2} , defined by the rule

g(x)= 2x
x3

is a onetoone function (injective) function, which is onto. Find the inverse function

g1 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 onetoone and
onto function. Find the rule f1(x) of the inverse function f1