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Domain / range of a function

  1. Oct 29, 2004 #1
    I'm a little confused here. I have a function: f(x) = x^2 + 3
    If the domain of the function is

    -3 <= x <= 3

    I need to find the range...

    I think that the answer is 3 <= y <= 12... that can be found out by drawing the graph. But I want to find the range algebraically. It says in my text book to turn the 'x' in the middle of the domain shown above into the respective f(x) equation (x^2)+3... and performing whatever action on x on the other sides of the inequality. This works in other functions, but I think I'm doing something wrong with this one:

    1.-3 <= x <= 3
    2.9 <= x^2 <= 9
    3.12 <= x^2 + 3 <= 12

    I think that because in step 1 I multiplied one value by (-3) I need to flip the greater than/equal to sign(s), but not sure how to do it... as you can see, by step 3, the wrong answer is reached. Can anyone show me my mistake? Thanks!
  2. jcsd
  3. Oct 29, 2004 #2

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    Hi preet - just curious...
    how did you come up with that domain? what's the problem with x = -4 or x = 4?x = -5 or x = 5? etc?
  4. Oct 29, 2004 #3
    The method you used only works if f(x) has a true inverse function which in this case it doesn't. This is because more than one x value exists for every y which is untrue of truely invertible functions. The function f(x) where x is lies on [0, [tex]\infty[/tex]) does have a true inverse. Take the domain to be 0<=x<=3 and then you can apply the rules you used.
    Last edited by a moderator: Oct 29, 2004
  5. Oct 29, 2004 #4

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    ah, nevermind.. preet must be referring to a restricted domain.
  6. Oct 29, 2004 #5


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    To go from an inequality like a<b to a^2<b^2 you need the additional assumption that a and b are non-negative 0<=a<b. Otherwise it may not be true, -4<3 but (-4)^2>3^2.

    You can go a couple ways, do 2 cases, one where x<=0 and one where x>0. Or you can recognize that -3<=x<=3 is equivalent to |x|<=3, and work with 0<=|x|<=3.
    Last edited: Oct 29, 2004
  7. Oct 30, 2004 #6
    The domain was given... anyways... I understand now. Thanks!
  8. Oct 31, 2004 #7
    I have another question related to functions... I thought I'd post it in here instead of starting a new thread...

    A graph of y=g(x) is as follows:
    (-4, 3), (-1, -2), (2, -2), (3, 1), (6, 1)

    the graph doesn't show a relation... it just shows a function... and the graph isn't a plot, so it isn't discrete.

    f(x) = -g (1/2x - 3) + 1

    Anyways, the questions are:
    a) Draw an input/output diagram for f.
    b) Determine the domain and range of f.
    c) Graph 'f'

    All I can do so far is
    a) [ add 6 ] + [ multiply by 2 ] + [ g ] + [ multiply by -1 ] + [ add 2 ]
    b) I don't know how to figure this out... the fact that function g is in function f is really confusing me...
    c) I don't know how to do this either... I was thinking that I have to apply the transformations in the input output diagram, but I want to figure out part b first.

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