What is the range and one-to-one condition for functions f, h, and g?

[allinurl]

Homework Statement

f : R+ → R+ defined by f(x) = 2x
h : R+ → N defined by h(x) = ⌊x⌋ (the largest integer ≤ x)
g : 2ᴿ → 2ᴿ defined by g(A) = A ∪ [0,1]

Homework Equations

The Attempt at a Solution

I know that every element of the domain has to correspond to exactly one element on the co-domain {range?}. But not sure how to determine if it satisfies the condition of one-to-one and its range.

 

[Office_Shredder]

I know that every element of the domain has to correspond to exactly one element on the co-domain {range?}

Ok, that's the definition of a function. The definition of one-to-one is that every element in the range corresponds to exactly one element in the domain. For example, in number 2. Is there more than one x such that h(x)=1?

 

[allinurl]

I would say that there is only one which is x=1. If I always plug the same number to the x1 and x2, the answer seems to be always same. for example,

if h(x₁) = h(x₂), then ⌊x₁⌋ = ⌊x₂⌋, so x₁ = x₂ This is why I'm confused.

Am I correct?

 

[Office_Shredder]

⌊1.2⌋=⌊1.1⌋?

 

[allinurl]

no, ⌊1.2⌋ is not equal to ⌊1.1⌋, but if so it means ⌊x₁⌋ is not equal to ⌊x₂⌋. In definition, the function if injective if a≠ b, then f(a)≠ f(b). So, is it one-to-one function?

 

[Office_Shredder]

I'm confused. What are ⌊1.1⌋ and ⌊1.2⌋ then? I would have though they're both 1

 

[Elucidus]

What is the greatest integer less than or equal to 1.2?
What is the greatest integer less than or equal to 1.1?

These are unequal?

--Elucidus

 

[allinurl]

oh~ok. I got that now. If so, in this case, ⌊x₁⌋ ≠ ⌊x₂⌋ but h(x₁) = h(x₂). I can say it is not one-to-one function. Correct?
I also have to say its range, how can I put?

And also, I have a problem to define A ∪ [0,1]. I know [0,1] means the set [xɛR | 0 ≤x ≤1]. But then, I'm not sure what I have to do..

 

[Office_Shredder]

oh~ok. I got that now. If so, in this case, ⌊x₁⌋ ≠ ⌊x₂⌋ but h(x₁) = h(x₂). I can say it is not one-to-one function. Correct?
I also have to say its range, how can I put?

h(x₁) = h(x₂)

this is equivalent to
⌊x₁⌋ = ⌊x₂⌋

Since h(x) = ⌊x⌋

I think you need to review the definition of greatest integer less than or equal to.

The same thing for the third part... it looks like you're unclear with concepts that your class is assuming you're familiar with. I highly recommend that you find out when you can go see your teacher for out of class help to catch you up to speed with the rest of the class