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It looks to me like they are all linearly independent but that is coming back as incorrect.

What is the general procedure for doing this anyway, I presume it is setting $c_1f(x) + c_2g(x) = 0$ and trying to find out if there are any non trivial solutions for c1 and c2...that is what I did but in every case I found c1 and c2 to be zero.

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For example the second case : they are linear dependent.Because one is equal to e^-9 multipliying the other for all x.

First case and the last : in the interval x>0 they are linear dependent, also in the interval with x<0, but there is no constant for all R, so in R they are linear independent. The third case are independent. You can suppose that one is equal to the other multiplied by a constant and get a contradiction. Or go to the definition, substitute two values of x, form a system , find that the only scalars are 0, and then they are the unique scalars for all x.

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All but 2. are linearly independent. The second pair of functions is linearly dependent, as $$g(x)=e^{3(x-3)}=e^{3x-9}=e^{-9}e^{3x}=e^{-9}f(x)$$ so $e^{-9}f(x)+(-1)g(x)=0$.

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