`f(x) = {{:(x^(2) sin.(pix)/(2)",",|x|lt 1),(x|x|", ",|x| ge l):}`, then f(x) is

Options

A. an even function
B. an odd function
C. a periodic function
D. none of these

Answer

B

Explanation

We have
`f(x)={{:(-x^(2)","if x le-1),(x^(2)sin""(pix)/(2)"," if -1 lt x lt 1),(x^(2)"," if x gt1):}`
For ` x in (-1,1)`, we have
`f(-x)=(-x)^(2) sin((pix)/(2))=-x^(2) sin""(pix)/(2)=-f(x)`
Let ` x in [1,oo).` Then ,`x=1+k ge0`
Clearly , `-x=-1-k in (-oo,-1]`.
Now , `f(-x)=f(-1-k)=-(-1-k)^(2)=-(1+k)^(2)`
and ` ,f(x)=(1+k)^(2)` .
`:. f(-x)=-f(x)` for all ` x in (-oo,1] cup [1,oo)`
Thus , `f(-x)=-f(x)` for all ` x in R `
Hence, f(x) is an odd functions.