**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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