For the following functions write the piecewise definition and draw the graph
(i) `f(x)=" sgn "(log_(e)x) " (ii) "f(x) = " sgn "(sin x)`

Explanation

(i) `f(x)=" sgn "(log_(e)x) `
`={(-1",",log_(e)x gt 0),(0",", log_(e)x=0),(1",",log_(e)x gt 0):}`
`= {(-1",",x in (0,1)),(0",",x=1),(1",", x in (1,oo)):}`

(ii) `f(x)=" sgn "(sinx)={(-1",",sinx gt 0),(0",", sinx=0),(1",",sinx gt 0):}`
`= {(-1",",x in ((2n+1)pi,(2n+2)pi)),(0",",x=n pi","n in Z),(1",", x in (2n pi,(2n+1) pi)):}`
Graphs of `y=sinx " and " y=" sgn "(sinx)` are as shown in the following figure.