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Question

Let g left parenthesis x right parenthesis equals 1 plus x minus left square bracket x right square bracket text  and  end text f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left left columnspacing 1em end attributes row cell negative 1 comma      text  if  end text x less than 0 end cell row cell 0      text  if  end text x equals 0 text  then  end text fraction numerator f left parenthesis g left parenthesis 2009 right parenthesis right parenthesis over denominator g left parenthesis f left parenthesis 2009 right parenthesis right parenthesis end fraction equals end cell row cell 1 comma      text  if  end text x greater than 0 end cell end table close

  1. x
  2. 1
  3. f left parenthesis x right parenthesis
  4. g left parenthesis x right parenthesis

Hint:

To find the given composite function we can first find each function separately and then divide the two. To solve f(g(x)) first we need to put the value of x in g(x) then the value obtained will be used in f(x). Same way we can fnd g(f(x)).

The correct answer is: 1


    For g left parenthesis x right parenthesis equals 1 plus x minus left square bracket x right square bracket text   and   end text f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left left columnspacing 1em end attributes row cell negative 1 comma      text  if  end text x less than 0 end cell row cell 0      text  if  end text x equals 0 end cell row cell 1 comma      text  if  end text x greater than 0 end cell end table close
    f(g(x))=f(1+x-open square brackets x close square brackets)
    f(g(2009))= f (1+2009-open square brackets 2009 close square brackets)
    = f (1+2009-2009)
    = f (1)
    =1  (as x is greater than 0)
    g(f(x))=g(f(2009))
    = g(1)     (f(2009)=0 as x is greater than 0)
    =1+1-open square brackets 1 close square brackets
    =2+1
    =1
    S o comma space fraction numerator f left parenthesis g left parenthesis 2009 right parenthesis right parenthesis over denominator g left parenthesis f left parenthesis 2009 right parenthesis right parenthesis end fraction equals 1 over 1 equals 1

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