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To find out if a function is even, odd, or neither, all one needs to do is replace the variable x with its opposite, -x. If the function remains exactly the same, it's an even function. If the function becomes a complete negative of itself, it's an odd function. If neither happens, then it's neither an even or odd function.
Now the simplest way to find out what this function is would be to just plot it out and look at its symmetry over the y-axis, but this can sometimes be difficult, depending on the function, so let's do it the algebraic way.
Our function is f(x) = x^3 + x^2 + x +1
Replace x with -x:
f(-x) = (-x)^3 + (-x)^2 + (-x) +1 = -x^3 + x^2 - x +1
Odd powers of a negative value always return another negative value and even powers always return a positive value. The function we got is obviously not the same as the original, so the function is not even, and we can't pull out an overall negative sign to return to our original function, so it's not odd either. This only leaves us with the option of neither.
Therefore our function is neither even nor odd.
I hope this explanation helped you better understand how to determine the sign of functions.
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