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Math Help - Help with continuous function.

  1. #1
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    Help with continuous function.

    Hello guys,
    I'm trying to understand this problem but having a hard time. Any help to point me in the right direction will be appreciated thanks

    For what value(s) of the constant c is the following function continuous on (−∞,∞)?



    f(x) = cx+1, x (less then or equal) 2 and cx^2, x > 2

    Sorry about the format. It's supposed to be set up to look like a piecewise function but i can't seem to get it like that
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  2. #2
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    Quote Originally Posted by Afterme View Post
    Hello guys,
    I'm trying to understand this problem but having a hard time. Any help to point me in the right direction will be appreciated thanks

    For what value(s) of the constant c is the following function continuous on (−∞,∞)?



    f(x) = cx+1, x (less then or equal) 2 and cx^2, x > 2

    Sorry about the format. It's supposed to be set up to look like a piecewise function but i can't seem to get it like that
     f\left( x \right) = \left\{ \begin{gathered}<br />
  cx + 1,{\text{   }}x \leqslant 2 \hfill \\<br />
  cx^2 {\text{       }}\;\;\;\;\;x > 2 \hfill \\ <br />
\end{gathered}  \right. \hfill \\

     {\text{For  }}f\left( x \right){\text{ to be continuous in }}\left( { - \infty ,\infty } \right){\text{, it should be continuous at }}x = 2. \hfill \\

      {\text{Left hand limit,}} = \mathop {\lim }\limits_{x \to 2 - } f\left( x \right) = \mathop {\lim }\limits_{x \to 2 - } \left( {cx + 1} \right) = 2c + 1 \hfill \\

     {\text{Right hand limit,}} = \mathop {\lim }\limits_{x \to 2 + } f\left( x \right) = \mathop {\lim }\limits_{x \to 2 + } \left( {cx^2 } \right) = 4c \hfill \\

     f\left( 2 \right) = 2c + 1 \hfill \\

    {\text{For  }}f\left( x \right){\text{ to be continuous at }}x = 2,{\text{ }} \Rightarrow \mathop {\lim }\limits_{x \to 2 - } f\left( x \right) = \mathop {\lim }\limits_{x \to 2 + } f\left( x \right) = f\left( 2 \right) \hfill \\

     \Rightarrow 2c + 1 = 4c \hfill \\

      \Rightarrow c = \frac{1}<br />
{2} \hfill \\
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  3. #3
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    You say for F(x) to be continuous, it has to be continuous at X = 2. Why though. I'm a bit confused on that part.
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  4. #4
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    Quote Originally Posted by Afterme View Post
    You say for F(x) to be continuous, it has to be continuous at X = 2. Why though.
    Do you know what it means for a function on the real numbers is continuous, period?
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  5. #5
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    For a function to be continuous the "Right hand limit" and "Left hand limit" must be approaching the same number for any give number.

    Both of those functions in the piecewise are continuous from negative infiniti to positive infiniti. But since at x=2 where the piecewise function 'switches functions' the two functions are not approaching the same number from the left and the right.

    So you must plug in a value for 'c' that will make them approach the same number. You do this by setting the two equations equal to each other, so at that point both parts of the piecewise equal the same number, and are therefore, (in this case at least), approaching the same number.
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  6. #6
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    Thanks fogel that cleared it up.
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