Let `f(x)={1+x ,0 lt x leq 2 and 3-x ,2 lt x leq 3` Determine the composite function `g(x) = f(f(x))` & hence find the point of discontinuity of g. if any.

Let `f(x)={1+x ,0 lt x leq 2 and 3-x ,2 lt x leq 3` Determine the comp

1.	Let `f(x)={1+x ,0 lt x leq 2 and 3-x ,2 lt x leq 3` Determine the composite function `g(x) = f(f(x))` & hence find the point of discontinuity of g. if any.
Answer» Given, `f(x) = {{:(1+x", "0 le x le 2),(3-x", " 2 lt x le 3):}` `:." "fof(x) = f[f(x)]={{:(1+f(x)", " 0 le f(x) le 2),(3-f(x)", "2ltf(x)le3):}` `= fof = {{:(1+f(x)" ,"0le f(x) le 1),(1+f(x)", "1ltf(x)le2),(3-f(x)", "2ltf(x) le 3):}={{:(1+(3-x)", "2lt xle 3),(1+(1+x)", " 0 le x le 1 ),(3-(1+x)", " 1 lt x le 2 ):}` `rArr (fof) (x) = {{:(4-x", " 2 lt x le 3),(2+x", " 0 le x le 1),(2-x", " 1 lt x le 2 ):}` Now, RHL (at x = 2) = 2 and LHL (at x = 2) = 0 Also, RHL (at x = 1) = 1 and LHL (at x =1) = 3 Therefore, f(x) is discontinuous at x = 1,2 ` :. ` f[f(x)] is discontinuous at x = {1, 2}.