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Homework Help: Range of a composite function

  1. Apr 15, 2010 #1
    1. The problem statement, all variables and given/known data

    The sets A and B are defined respectively by

    [tex]A={x\in R : 0\leq x\leq 1}[/tex]

    [tex]B={x\in R : 1\leq x\leq 2}[/tex]

    and the functions f and g are defined respectively by

    [tex]f(x)=x^2-2x+2[/tex]

    [tex]g(x)=\frac{x+2}{x-1}[/tex]

    where f(A)=B , g(B)=C with C as the range of the function g .

    Find the range of the composite function gf(A)=C

    2. Relevant equations



    3. The attempt at a solution

    [tex]gf(x)=1+\frac{3}{(x-1)^2}[/tex] with domain [0,1)

    so the range is (1, infinity)
     
  2. jcsd
  3. Apr 15, 2010 #2

    Cyosis

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    Homework Helper

    That doesn't seem right. Are you sure you have defined your sets A and B correctly? As it is now the domain of B includes 1, but 1 is not a valid value, because g(1) is undefined.

    Secondly if you plug a function f that sends A to B into another function g then g should take values from B as input. You on the other hand have g taking values from A as input.

    Lastly if the domain of g o f is [0,1) then the range you found is certainly wrong. g o f is a function that increases in the interval [0,1) so the minimum value of the range cannot be 1 since (g o f)(0)>1.
     
  4. Apr 15, 2010 #3
    ok , i see my mistake , so the correct range should be [4 , infinity) ? But the answer given is 2/3<=x<=4/3
     
  5. Apr 15, 2010 #4

    Cyosis

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    Homework Helper

    You didn't answer all my questions:

    Secondly if 2/3 is in the range of g o f then 2/3=1+3/(x-1)^2 has a solution for x which is in the domain of g o f. Check that this is not the case.
     
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