fx x1x gx x1x2 Find a gfx b ffx c ggxSolutiongiven fx x1x gx

f(x)= x+(1/x)    g(x)= (x+1)/(x+2)

Find

a) g(f(x)

b) f(f(x)

c) g(g(x)

Solution

given

f(x)= x+(1/x)    g(x)= (x+1)/(x+2)

a) g(f(x) = g[x + (1/x) ]

             = [ x+(1/x) +1 ] / [ x+(1/x) +2]

            = [ x^2 + x +1 ] / [x^2 +2x+1 ]

           = [x^2 +x +1] / (x+1)^2

b) f(f(x) = f(x+(1/x)

             = x+(1/x) + 1 / (x+1/x)

             = x+(1/x) + x/(x^2+1)

c) g(g(x) = g(x+1/x+2)

               = [(x+1 / x+2) +1 ] / [(x+1/x+2 )+ 3]

              = (x+1 +x+2) / (x+1 +3(x+2) )

             = 2x+3 / (x+1 +3x+6)

             = 2x+3 / 4x+7