17. Even-odd decompositions
b. If f(x) = f_E(x) + f_O(x)...

Question

17. Even-odd decompositions

b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that

f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2

Accepted Answer

Welcome back, everyone. Let HX be equal to H E X plus H O X, where H E X is even and HO X is odd. Which formula correctly gives H E X in terms of HX and H negative X A H X plus H negative X divided by 4. B H X minus H X divided by 2, C H X minus H X divided by 2, and D H X plus H negative X divided by 2. So for this problem, we know that H E X is even, meaning H E of negative X must be equal to H E X. And then HO is odd, so HO of negative X must be equal to negative HO X. So these are the initial conditions. And now, if we consider the expression of H X, we're going to evaluate H of negative X, which is going to be equal to H E of negative X plus H O negative X. We can now express each of these functions in terms of their symmetry. So H E of negative axis, H E X. And we are adding HO of negative X, meaning we're basically subtracting HO X. Now from here, what we can do is simply add H of X to both sides of this expression. So we're going to get H of negative X plus H of X. It's equal to HE. Of X. Minus HO of X. Las H E. Of X. Plus HO of X. And now what we can do is simply understand that on the left hand side, we have H of negative X plus H X, so nothing changes, right? And on the right hand side, this sum is basically equal to 2 H E X. Why is that the case? Well, we can cancel out HO of X, right? So let's go ahead. And cancel out these terms. And now we want to isolate H E X. So what we can do is simply divide both sides by 2, right? Meaning H E X is equal to H of negative X plus. H X. Let's ensure that we have enough space to write the final answer. So H E X equals H of negative X plus H X. Divided by 2, which corresponds to the answer choice D. Thank you for watching.

17. Even-odd decompositionsb. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show thatf_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2

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17. Even-odd decompositions
b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that
f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2