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Problem 76 Hard Difficulty

Determine whether $ f $ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$ f(x) = x | x | $

Answer

Odd function.

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Video Transcript

all right. We want to determine if this function is even on or neither. And so remember that for odd functions, opposite X values have opposite. Why values? And so the grafts have origin, symmetry and for even functions opposite X values have the same y value. And so the grafts have y axis symmetry. So what we want to do is determine if f of the opposite of X is going to be the same as or the opposite of, or neither compared to f of X. So we substitute the opposite of X in the function, and we get the opposite of X times the absolute value of the opposite of X. Now realize that the absolute value of the opposite of X is just the same thing as the absolute value of X X, and the opposite will be the same distance from zero, so we can simplify it. Now that looks like the opposite of what we originally had. So we've just found that f of the opposite of X is equal to the opposite of F of X, which means this function is odd. Now we can grab it on a calculator and see the origin symmetry. So we type it into the calculator and we look at the graph and there you see that origin symmetry.