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Range of a function

  1. Nov 15, 2009 #1
    Help needed.

    1. The problem statement, all variables and given/known data
    Functions g and h are defined as follows:
    g : x → 1 + x x ∈ R
    h : x → x² + 2x x ∈ R

    Find i.) the ranges of g and h,

    ii.) the composite functions h ° g and g ° h, stating their ranges.
    Not sure how this is to be done help needed, please.

    2. The attempt at a solution

    i.) range of g => R = {y : y ∈ R}

    1 + x = 0
    x = -1
    1 - 2 = -1
    -b/2a = -2/2 = -1
    range of h => R = {y : y ≥ - 1, y ∈ R}
     
  2. jcsd
  3. Nov 15, 2009 #2

    CompuChip

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    The way to do this explicitly, would be to simply give the formula for h o g and g o h, which will give you two quadratic functions again.

    But you can also find it by argument. For example, consider h o g. You take some x and apply g. What are the possible values y that you get. Then apply h. What can y map to?
     
  4. Nov 15, 2009 #3

    HallsofIvy

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    h(x)= [itex]x^2+ 2x= x^2+ 2x+ 1- 1= (x+1)^2- 1[/itex]
    I presume that is why you were looking at "1+ x= 0"!

    Now, g(x) can be any number so h(g(x)) can be what?

    h(x) must be larger than or equal to 1 so g(h(x)) can be what?
     
  5. Nov 15, 2009 #4
    Not sure how much of a difference this is:
    g : x |→ 1 + x x ∈ R
    h : x |→ x² + 2x x ∈ R

    This would be the range of h(x)=> [itex] x^2 + 2x = x^2 + 2x + 1 - 1 = (x + 1)^2 - 1[/itex]
    and my range for g is correct?

    Now, g(x) can be any number so h(g(x)) can be what?
    h(g(x)) can be any number. R = {y: x ∈ R}

    h(x) must be larger than or equal to 1 so g(h(x)) can be what?
    g(h(x)) can be larger than or equal to 1. R = {y : ≥ - 1, y ∈ R}
     
    Last edited: Nov 15, 2009
  6. Nov 15, 2009 #5

    CompuChip

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    Your result for h(x) already was correct (it's {y | y ≥ -1}).
    Halls was just pointing out that h(x) = g(x)2 - 1, I suppose (which you could also have used to obtain the same result).

    Note that h(g(x)) means you are first evaluating g on x. This can give you any number, which you plug into h...
     
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