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# Calculate some values for some even and odd functions with given properties

Let be an odd function, integrable everywhere, with Let be an even function, integrable everywhere, with Prove the following:

1. for all ;
2. ;
3. .

1. Proof. We compute using the given properties, 2. Again, we compute using the given properties of and , 3. Finally, 1. AudioRebel says:

a) g(x) = f(x + 5) => given
g(-x) = f(-x + 5) =>
g(x) = f( – [x – 5] ) => g(x) is even
g(x) = -f(x – 5) => f(x) is odd
-g(x) = f(x – 5)

2. Andres says:

I think the book says that f is odd and g is even, so there is a mistake above.

3. Anonymous says:

in b , after the third = sign shouldnt there be a minus sign ?

• Sebastian says:

He did two steps in one there.

• Mihajlo says:

Solutions here a) and b) are wrong, because he used that g is odd (and not even, as is asked in the book).

But the correct proofs are very similar to what he did, you just need to use integral properties in a) as well.