Mapping Functions: Is ∅ an Isomorphism?

[taylor81792]

Homework Statement

Let F be the set of all functions f mapping ℝ into ℝ that have derivatives of all orders. Determine whether the given map ∅ is an isomorphism of the first binary structure with the second.

{F,+} with {ℝ,+} where ∅(f)=f'(0)

Homework Equations

None

The Attempt at a Solution

If I have to determine it is isomorphism, I have to prove it is onto and 1-1, but I'm not sure how to do that with ∅(f)=f'(0)

 

[SteveL27]

Homework Statement

Let F be the set of all functions f mapping ℝ into ℝ that have derivatives of all orders. Determine whether the given map ∅ is an isomorphism of the first binary structure with the second.

{F,+} with {ℝ,+} where ∅(f)=f'(0)

Homework Equations

None

The Attempt at a Solution

If I have to determine it is isomorphism, I have to prove it is onto and 1-1, but I'm not sure how to do that with ∅(f)=f'(0)

Is ∅ injective? (Same as asking if it's one-to-one). That is, does ∅ always send different functions to different reals?

 

[Deveno]

i suggest taking a handful of functions, and calculating f'(0) for each of them.

possible candidates:

f(x) = xⁿ (don't forget the special cases n = 0, and n = 1)
f(x) = ax + b
f(x) = sin(x)
f(x) = cos(x)

do any of these have the property that θ(f₁) = θ(f₂) but f₁ ≠ f₂?

 

[taylor81792]

so I would try doing f(x)=X^n. and then get 0^0 and 0^1? so the answers would be 1 and 0. I'm still a little bit confused

 

[Dick]

so I would try doing f(x)=X^n. and then get 0^0 and 0^1? so the answers would be 1 and 0. I'm still a little bit confused

If that's confusing you, start with f(x)=x and f(x)=sin(x).

 

[Mark44]

so I would try doing f(x)=X^n. and then get 0^0 and 0^1? so the answers would be 1 and 0. I'm still a little bit confused

If n = 0, then f(x) = 1, so f(0) = 1.
If n = 1, then f(x) = x, so f(0) = 0.

 

[Deveno]

unless i am mistaken, the problem is asking for the derivative of f at 0, not f(0).