Final answer:
The odd extension f_o(x) of the function f(x)=1-x on [0, 2], is defined as f_o(x) = f(x) for 0 ≤ x ≤ 2 and f_o(x) = -f(-x) for -2 ≤ x < 0. The graph is symmetric about the origin and to graph it over three periods, you reflect and repeat the original function maintaining odd symmetry.
Step-by-step explanation:
Odd Extension of a Function
To define the odd extension of the function f(x) = 1 - x on the given interval [0, 2], we need to extend f in such a way that fo(-x) = -fo(x). The piecewise definition of the odd extension, fo(x), will be:
- fo(x) = f(x) for 0 ≤ x ≤ 2
- fo(x) = -f(-x) for -2 ≤ x < 0
To sketch the graph of fo, we reflect the graph of f on the y-axis and invert it. fo will thus be symmetric with respect to the origin. To sketch the graph over three periods, simply repeat the piecewise graph of fo to the left and right as needed, maintaining the odd symmetry.
Graph of f(x) = 1 - x
The original function f(x) = 1 - x on [0, 2] is a straight line that goes down from (0,1) to (2,-1). The odd extension fo will mirror this line across the y-axis from (-2,1) to (0,-1) for negative values of x.
Remember, for an function, the integral over all space is zero due to the symmetry around the origin. However, the original function given on [0, 2] is not an odd function itself but can be extended to create an odd function.