Give an example of a function that is neither even nor odd and explain algebraically why it is neither even nor odd.

So, first off, it's important to understand what exactly an even and an odd function actually is.

An even function can be described like this:

f (-x) = f (x)

This is just saying that if you plug in "-x" into your function, you'll recieve the same function that you started with when you simplified.

An example of this is f (x) = x^2. If I plug -x into f (x) I will still get x^2 after I simplify. If this doesn't make sense, let me know and I can do the actual work for this.

An odd function can be described as:

f (-x) = -f (x)

This means that when you plug "-x" into your function, you get something that's equal to if you multiplied your function by a -1.

An example of this is f (x) = x^3. If I plug -x into this function, I get (-x)^3, which can simplified to -x^3. This is t S as if I multiplied x^3 by -1.

So! how do we find a function that is neither even nor odd? Well, to do this, we have to understand what exactly makes those t above examples even and odd. Specifically, it is their EXPONENTS.

Anything squared is just a positive number, so putting a negative number in an even exponent is the same as if you put a positive number under an exponent. So, if we added an variable to the function x^2 that didn't have a even exponent, we would no longer be an even function. You could apply this same logic to odd functions.

An odd function is odd because it contains an odd exponent, like 3. If we add a variable that doesn't have an odd exponent, it will no longer be odd. Of course, there are other way to develop non odd or even functions, but I believe this to be the most straightforward one.

I think from here you can probably come up with one, but if not, I'd be happy to help more in the comments. Hope this makes sense! :)