# Mapping Functions

1. Feb 22, 2012

### taylor81792

1. The problem statement, all variables and given/known data
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)

2. Relevant equations

None

3. 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)

2. Feb 22, 2012

### SteveL27

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

3. Feb 22, 2012

### 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?

4. Feb 22, 2012

### 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

5. Feb 22, 2012

### Dick

6. Feb 22, 2012

### Staff: Mentor

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

7. Feb 22, 2012

### Deveno

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