# Determine whether eacg function is odd, even, or neither. a) y = x^3sinx b) y= x + sin^2 x <--- the x is not part of the exponent for the sin^2

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### 1 Answer

A function is even if f(-x) = f(x) and odd if f(-x) = -f(x) otherwise it is neither even or odd. `x^2` is even `x^3` is odd, cos(x) is even, sin(x) is odd.

a) `f(x)=x^3sin(x)`

`f(-x) = (-x)^3 sin(-x) = -x^3 (-sin(x)) = x^3sin(x)` so `y = x^3sin(x)` is even not odd, because `f(-x) = f(x)`

b) `f(x) = x + sin^2(x)`

`f(-x) = (-x) + sin^2(-x) = -x + (sin(-x))^2 = -x + (-sin(x))^2 = -x + sin^2(x)` which is neither even nor odd because `f(-x) != f(x)` and `f(-x) != -f(x)`