16 SYMMETRYTest for symmetry: Even and odd functions LET THIS BE THE RIGHT-HAND SIDE of the graph of a function: We will now draw the left-hand side -- so that the graph will be symmetrical with respect to the y-axis: In this case, f(−x) = f(x). The height of the curve at −x is equal to the height of the curve at x -- for every x in the domain of f. Again, let this be the right-hand side: We will now draw the left-hand side -- so that the graph will be symmetrical with respect to the origin: Every point on the right-hand side is reflected through the origin. In this case, f(−x) = −f(x). The value of f at −x is the negative of the value at x. (A reflection through the origin is equivalent to a reflection about the y-axis, followed by a reflection about the x-axis.) Test for symmetry: Even and odd functions Symmetry, then, depends on the behavior of f(x) on the other side of the y-axis -- at minus-x : f(−x). Here is the test: If f(−x) = f(x), then the graph of f(x) is symmetrical with respect to the y-axis. To see the answer, pass your mouse over the colored area. If f(−x) = −f(x), then the graph of f(x) is symmetrical with respect to the origin. A function symmetrical with respect to the y-axis is called an even function. A function that is symmetrical with respect to the origin is called an odd function. Example 1. Test this function for symmetry: f(x) = x^{4} + x^{2} + 3 Solution. We must look at f(−x):
Since f(−x) = f(x), this function is symmetrical with respect to the y-axis. It is an even function. Example 2. Test this function for symmetry: f(x) = x^{5} + x^{3} + x Solution. Again, we must look at f(−x):
Since f(−x) = −f(x), this function is symmetrical with respect to the origin. It is an odd function. Problem. Test each of the following for symmetry. Is f(x) even, odd, or neither? a) f(x) = x^{3} + x^{2} + x + 1 Answer. Neither, because f(−x) ≠ f(x) , and f(−x) ≠ −f(x). b) f(x) = 2x^{3} − 4x Answer. f(x) is odd -- it is symmetrical with respect to the origin -- because f(−x) = −f(x). c) f(x) = 7x^{2} − 11 Answer. f(x) is even -- it is symmetrical with respect to the y-axis -- because f(−x) = f(x). 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. In trigonometry, y = sin x is an odd function, while y = cos x is even. Therefore, the issue is the test of f(−x). Please make a donation to keep TheMathPage online. Copyright © 2014 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |