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Function Notation: Even and Odd (page 3 of 3) Sections: Introduction & Evaluating at a number, Evaluating at a variable, Even and odd functions You may be asked to "determine algebraically" whether a function is even or odd. To do this, you take the function and plug –x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f(–x) = f(x), so all of the signs are the same), then the function is even. If you end up with the exact opposite of what you started with (that is, if f(–x) = –f(x), so all of the "plus" signs become "minus" signs, and vice versa), then the function is odd. In all other cases, the function is "neither even nor odd".
But the question asks me to make the determination algebraically, so I'll plug –x in for x, and simplify:
f(–x)
= 2(–x)^{3} – 4(–x)
My final expression is the exact opposite of what I started with, by which I mean that the sign on each term has been changed to its opposite, just as if I'd multiplied through by –1: –f(x)
= –1[f(x)]
This means that f(x) is odd.
You may find it helpful, when answering this "even or odd" type of question, to write down f(x) and –f(x) explicitly, and then compare them to whatever you get for f(–x). This can help you make a confident determination of the correct answer. You can use the Mathway widget below to practice figuring out if a function is even, odd, or neither. Try the entered exercise, or type in your own exercise. Then click the "paperairplane" button to compare your answer to Mathway's. (Or skip the widget and return to the index.)
(Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.) << Previous Top  1  2  3  Return to Index



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