Could somebody help me prove this math proof on one-to-one!! Please...Pts will be ginven out tomorrow night!?

Question

Plese someone help me with this proof...i need to know HOW to do it in two day!! Proofs are confusing to me...please help!!

Proof: Prove that if a funtion is increasing on [a,b] then f(x) is one-to-one...use definitions in your proof if possible.

Answer 1

If f(x) is everywhere increasing on [a,b] then for all
x_1, x_2 ε [a,b] with x_1 < x_2, then f(x_1) < f(x_2) (by the definition of increasing) and if f(x_1) = f(x_2) then it must be that x_1 = x_2 (again, by definition of increasing).

Thus f is 1-1 since each x ε [a,b] produces a unique f(x)

P.S. It would be a very good idea for you to work on proofs, because they are at the very heart of all mathematics ☺


Doug

Answer 2

If f(x1) = f(x2) then x1=x2 This is the definition of a one-to-one function.

A function is increasing if for all x1 < x2 then f(x1) < f(x2)

Proof by Contradiction:

Assume f(x) is increasing, and NOT one-to-one
Since f is not one-to-one, there exists, without loss of generality, x1 < x2 such that f(x1) = f(x2). But we know f(x1) < (fx2) since
f(x) is increasing. This is a contradiction. Thus f(x) must be one-to-one.

Answer 3

u got it wrong there. there is a difference in increasing and strictly increasing. an increasing function is not necessarily one to one, but a strictly increaing one is. and if u are feeling that damn proof difficult then how will you study calculus???