What is an odd function? A function f is an odd function if f(-x) = -f(x) for all x in the domain of f.
For example f(x) = x^{3} is an odd function
f(-2) = (-2)^{3} = -2 × -2 × -2 = -8 and f(2) = (2)^{3 }= 2 × 2 × 2 = 8
f(-2) = -8 = -f(2)
f(-1) = (-1)^{3} = -1 × -1 × -1 = -1 and f(1) = (1)^{3 }= 1 × 1 × 1 = 1
f(-1) = -1 = -f(1)
In general, let x represent any number in the domain of f.
Then, f(-x) = (-x)^{3} = -x × -x × -x = -x × x^{2} = -x^{3} and f(x) = x^{3}
f(-x) = -x^{3} = -f(x)
For f(2) = 8 and f(-2) = -8, the points are (2, 8) and (-2, -8)
For f(1) = 1 and f(-1) = -1, the points are (1, 1) and (-1, -1)
Notice that f(0) = 0^{3} = 0, so there is a single point (0,0)
Now, try to graph these five points on a coordinate system.
(2, 8), (-2, -8), (1, 1), (-1, -1), and (0, 0)
From the graph above, we can make the following two important observations:
f(x) = x^{3} + 1 is not an odd function.
f(-x) = -x^{3} + 1
f(-x) = -(x^{3} - 1)
Since x^{3} - 1 ≠ x^{3} + 1, f(-x) ≠ -f(x)
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