12 points Consider the function f defined on 0 fxxx1x22 x0

(12 points) Consider the function f defined on [0, ): f(x)=x(x1)(x2)2, x[0,).

(a) Extend f into an even function F such that it agrees with f on [0, ): Write down F(x) in function form, and also sketch its graph in the coordinate plane.

(b) Extend f into an odd function G such that it agrees with f on [0,): Write down G(x) in function form, and also sketch its graph.

Solution

In an even function f(-x)= f (x) , irrespective of x\'s value.

In an odd function f (-x)= -f (x)

a). To convert f (x) into an even function take mod value or absolute value of f (x)

F (x)= | x (x-1)(x-2)2 | here we can verify that it will satisfy for f [0, infinity ) , for x=1.5, f (x)= negative value, thus to convert f into even function F, we take the mod value.

b) f can\'t be converted to an odd function because as the criteria described above, f is defined only for positive values of x, hence x will never have a negative value.

For plotting the graph, replace f (x)=y or F (x)=y

Then for a 2D graph, substitute different values of x to get different values of y and accordingly plot.