Odd functions have what kind of symmetry

I want to talk about even and odd functions. First the definition. A function f is even if f of -x equals f of x for all x in the domain of f. That means that you can switch x for -x and get the same value. Now what kind of symmetry does that give us? Well the graph of an even function's always going to be symmetric with respect to y axis. Why is that? Well, if you remember our discussion of symmetry, of reflections, the graph of y equals f of -x. It is an odd function.

Test each of the following for symmetry. Is f x even, odd, or neither? Note: A polynomial will be an even function when all the exponents are even. A polynomial will be an odd function when all the exponents are odd. But there are even and odd functions that are not polynomials. So this is doing two flips.

So some of you might be noticing a pattern or think you might be on the verge of seeing a pattern that connects the words even and odd with the notions that we know from earlier in our mathematical lives. I've just shown you an even function where the exponent is an even number, and I've just showed you an odd function where the exponent is an odd number. Now, I encourage you to try out many, many more polynomials and try out the exponents, but it turns out that if you just have f of x is equal to, if you just have f of x is equal to x to the n, then this is going to be an even function if n is even, and it's going to an odd function if n is odd.

So that's one connection. Now, some of you are thinking, "Wait, but there seem to be a lot of functions "that are neither even nor odd. For example, if you just had the graph x squared plus two, this right over here is still going to be even. You're going to get back to itself. But if you had x minus two squared, which looks like this, x minus two, that would shift two to the right, it'll look like that.

That is no longer even. Because notice, if you flip it over the y-axis, you're no longer getting the same function. So it's not just the exponent. It also matters on the structure of the expression itself. If you have something very simple, like just x to the n, well, then that could be or that would be even or odd depending on what your n is.

Similarly, if we were to shift this f of x, if we were to even shift it up, it's no longer, it is no longer, so if this is x to the third, let's say, plus three, this is no longer odd. Because you flip it over once, you get right over there. You may find it helpful, when answering this "even or odd" type of question, to write down — f x explicitly, and then compare this 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 "paper-airplane" button to compare your answer to Mathway's. Please accept "preferences" cookies in order to enable this widget. Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.

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