anonymous one year ago let f and g are functions that are neither even nor odd. a)Create an example where f+g is even, b) f+g is odd, c)f.g is even, d) f.g is odd

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1. Loser66

I give you one case , b, as example, you do the rest, ok? Let $$f(x) = x+3$$ let check $$f(-x) = -x +3 \neq f(x) ~~hence,~~\text{f(x) is not even}$$ $$-f(x) = -x-3 \neq f(-x)~~hence,~~\text{f(x) is not odd, therefore f(x) is neither even nor odd}$$ Now, let $$g(x) = -2x-3$$, let check $$g(-x) = 2x-3\neq g(x),~~hence,~~g(x) \text{is not even}$$ $$-g(x) = 2x+3 \neq g(-x),~~hence~~\text{g(x) is not odd, hence g(x) is neither even nor odd}$$ Now, combine $$(f+g)(x) = f(x) +g(x) = x+3-2x-3 = -x$$ let check $$(f+g)(-x) = x \\(-(f+g)(x) = -(-x) =x$$ hence $$(f+g)(-x) = x=-(f+g)(x)$$, therefore $$(f+g)(x)$$is an odd function.