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Piecewise functions

  1. Sep 27, 2009 #1
    1. The problem statement, all variables and given/known data
    We started piecewise functions in calculus and I'm finding it rather difficult. This question has had me stumped for a while. I'm quite well at finding domain and range, however :).

    For what value of k is the following a continuous function

    2. Relevant equations

    [tex]f(x)= \frac {x-2}{\sqrt {7x+2} -\sqrt{6x+4}} [/tex] [tex] if x \geq {\frac{-2}{7}}
    and x \neq 2[/tex]
    f(x) = k, if x=2

    I hope this makes sense.
    3. The attempt at a solution

    The problem I am having is how I can find k - the constanst. I can't [tex]\frac {x-2}{\sqrt {7x+2} -\sqrt{6x+4}} = k [/tex] since k cannot equal to 2.
     
  2. jcsd
  3. Sep 27, 2009 #2
    You want to cancel out or get rid of the denominator and rewrite the expression in a way where you don't get undefined when x=2. Try multiplying by the conjugate of the denominator.
     
  4. Sep 27, 2009 #3
    I was thinking of just rationalizing it. But what do I do with the k then?
     
  5. Sep 27, 2009 #4
    If you graph f(x) on a graphing calculator, you would see that the graph clearly approaches a value as x gets close to 2. That is because when you plug x=2 into the function, you get 0/0 which usually, but not always, means that the function actually approaches a value as x gets close to the value that it can't be, the one that makes the function undefined. Once you find the value k that f(x) gets close to as x approaches 2, that will be the value to make the function continuous.

    Think of x(x-1)/(x-1) which is undefined at x=1 and its graph has a "hole" there. If I do some canceling, I get just x. Then I can let x=1 and see that the graph of x(x-1)/(x-1), which coincides with the graph of x, gets close to 1 as x approaches 1, just like the graph of x when x gets close to 1.
     
  6. Sep 27, 2009 #5
    So if I rationalize it, to remove the square roots, can I equate the 2? Then use x = -2/7?
     
  7. Sep 27, 2009 #6
    I rationalized it:

    [tex] \frac {x-2 \sqrt {7x+2} + x-2\sqrt{6x+4}}{x-2}[/tex]


    [tex] \sqrt {7x+2} + x-2\sqrt{6x+4} = k [/tex]

    Substituting x=2, I get k=4.
     
  8. Sep 28, 2009 #7
    k isn't 4. After multiplying by the conjugate of the denominator and simplifying, the x-2 should cancel; looks like you didn't distribute correctly.

    [tex]\frac{x - 2}{\sqrt{7x + 2} - \sqrt{6x + 4}} * \frac{\sqrt{7x + 2} + \sqrt{6x + 4}}{\sqrt{7x + 2} + \sqrt{6x + 4}} = \frac{(x - 2)(\sqrt{7x + 2} + \sqrt{6x + 4})}{7x + 2 - (6x + 4)} = \frac{(x - 2)(\sqrt{7x + 2} + \sqrt{6x + 4})}{x - 2}[/tex]
     
    Last edited: Sep 28, 2009
  9. Sep 28, 2009 #8
    Ohhh, you don't multiply the x-2 in to the conjugate? I understand. I got the x-2 on both the top and bottom, but I multiplied through. I see what I did wrong, thank you.
     
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