# Let # f(x) = |x-1|. # 1) Verify that # f(x) # is neither even nor odd. 2) Can # f(x) # be written as the sum of an even function and an odd function ? a) If so, exhibit a solution. Are there more solutions ? b) If not, prove that it is impossible.

##### 1 Answer

Let

If f were even, then

If f were odd, then

Observe that for x = 1

Since 0 is not equal to 2 or to -2, f is neither even nor odd.

Might f be written as

If that were true then

Replace x by -x.

Since g is even and h is odd, we have:

Putting statements 1 and 2 together, we see that

ADD THESE to obtain

This is indeed even, since

From statement 1

This is indeed odd, since