 Let me introduce one more idea about functions. A function f of x is even if f of x equals f of negative x for all x, and odd if f of x is equal to negative f of x for all x. So we can determine whether a function is even, odd, or neither. Definitions are the whole of mathematics. All else is commentary. Let's go ahead and bring in our definition of even and odd. And note that this requires us to compare f of x and f of negative x, so we'll compare f of x, f of negative x, and, if necessary, negative f of x. So we'll find f of negative x, and we see that this is equal to f of x, and so f of x is even. Or how about this function? We'll compare g of x, g of negative x, and negative g of x. So we'll find g of negative x, and simplify. And this is not g of x, so we'll find negative g of x, and simplify. And we see that g of negative x and negative g of x are the same thing, and so g of negative x is negative g of x, and g is odd. Or a function like this, we'll compare h of x and h of negative x, h of negative x is, and h of x is, negative h of x is, and h of negative x is neither of these, and so h of x is neither even nor odd. What about graphing? Suppose f is an even function. If some point hk is on the graph of y equals f of x, we have k equals f of h. But because f is an even function, we have f of h equals f of negative h. Equals means replaceable, since k equals f of h, we have k equals f of negative h. And so the point negative hk is also on the graph. But this means the graph is symmetric about the y-axis. Now suppose g is an odd function. Again, if hk is on the graph of y equals g of x, we have k equals g of h. But since g is an odd function, g of negative h is negative g of h. Equals means replaceable, and so g of negative h is negative k, and so negative h, negative k, is also on the graph. And this means the graph is symmetric about the origin. So for example, if we know that part of the graph of y equals f of x is shown, if f of x is an even function, we can sketch the remainder of the graph. So again, since f of x is even, then any point hk on the graph has k equal to f of h, so f of negative h is also equal to k. And so negative hk is also on the graph, and so the graph is symmetric about the y-axis.