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Notice that in order to have x-axis symmetry we have to violate the vertical line test. Therefore we no longer have a function. |
Function Symmetries
A function will not be symmetric with respect to the x-axis unless all of its points lie on the x-axis. (i.e. y = 0). Why? Well look what happens if we have x-axis symmetry:
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Notice that in order to have x-axis symmetry we have to violate the vertical line test. Therefore we no longer have a function. |
example: f(x) = x2 + 4, so f(-x) = (-x)2 + 4
= x2 + 4
= f(x)
A function f(x) is symmetric with respect to the origin if for every x in its domain, f(-x) = -f(x). This kind of function is called an odd function.
example: f(x) = x3 + x, so f(-x) = (-x)3 + (-x)
= -x3 - x
= -(x3 + x) = -f(x)
Notice that checking for symmetries of functions is easier than checking for symmetries in general, because we only have two basic symmetries to worry about and we can use just one test to check for both of them! (On the symmetries lecture, which you should have worked through before this one, you had three symmetries to check for and you had to do a different test for each one.)
Examples: Graphically determine whether each of the following functions are even, odd, or neither. Then back up your suspicions algebraically.
Click here for answers.
f(x) = 3x4 - 2x2 + 1 g(x) = 5x3 + 2x - 1 h(x) = x2/3