The Common Denominator: Determine if a function is even, odd, or neither

You may be asked to "determine algebraically" whether a function is even, odd or neither. To negotiate this problem, you simply make a substitution of -x (negative x) for x; simplify; and compare the terms indiviudally. Afterwards, if the functions are the same, that is f(x) = f(-x), then the function is even. However, if the functions are opposite, that is f(x) = - f(-x), then the function is odd. In all other cases, the function is neither. If we take a close look at the graph of the functions, we will see that the symmetry is illustrated simply by plotting the function, which may show symmetry with respect to either an axis (y-axis) or origin. In the examples here, the even functions are x-squared, |x|, and cos(x).  However, the odd functions are x (a linear function), x-cube, and sin (x).


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