Stitz-Zeager_College_Algebra_e-book

Consider the graph of the function f given on the

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Unformatted text preview: of f . • About the origin if and only if f (−x) = −f (x) for all x in the domain of f . For reasons which won’t become clear until we study polynomials, we call a function even if its graph is symmetric about the y -axis or odd if its graph is symmetric about the origin. Apart from a very specialized family of functions which are both even and odd,3 functions fall into one of three distinct categories: even, odd, or neither even nor odd. Example 1.7.3. Analytically determine if the following functions are even, odd, or neither even nor odd. Verify your result with a graphing calculator. 5 2 − x2 5x 2. g (x) = 2 − x2 5x 3. h(x) = 2 − x3 1. f (x) = 5x 2x − x3 x 5. j (x) = x2 − −1 100 4. i(x) = Solution. The first step in all of these problems is to replace x with −x and simplify. 2 3 Why are we so dismissive about symmetry about the x-axis for graphs of functions? Any ideas? 1.7 Graphs of Functions 67 1. f (x) = f (−x) = f (−x) = 5 2 − x2 5 2 − (−x)2 5 2 − x2 f (−x) = f (x) Hence, f is even. The graphing calculator furnishes...
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