Question

If f : R → R is defined by f(x) = 2x-3. Prove that f is a bijection and find its inverse.

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Answer 1

A N S W E R :

F  is bijective!

STEP BY STEP EXPLANATION :

Recall that  F:A→B  is a bijection only if  F  is

injective:  F(x)=F(y)⇒x=y , and

surjective:  ∀b∈B  there is some  a∈A  such that  F(a)=b .

So, is  f  an injection?

Take  x,y∈IR  and assume that  f(x)=f(y) . Therefore  2x−3=2y−3 . We can cancel out the  3  and divide by  2 , then we get  x=y .

Is  f  a surjection?

This is a trivial case since the two sets of the function are the set of real numbers. Every  x∈IR  is bound to a value defined by  2x−3 .

Therefore:  F  is bijective!

Answer 2

Answer:

How many numbers are there between 100 and 500 with the digits 0, 1, 2, 3, 4, 5 if

(i) repetition of digits allowed

(ii) the repetition of digits is not allowed