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Math Help - Continuity Functions

  1. #1
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    Continuity Functions

    I'm at a loss on where to start with either of these problems. It seems you will use the same basic methods of solving each so I will list each of them.

    1.) For what value of the constant is the function continuous on where


    2.) For what value of is the function defined below continuous on ?


    Thanks for help in advance.
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  2. #2
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    Hello, jwebb19!

    1.) For what value of the constant c is the function f continuous?

    . . f(t) \;=\;\begin{Bmatrix}t^2-c,& t < 6 \\ ct + 3, & t \geq 6\end{Bmatrix}

    To the left of t = 6, the graph is a parabola -- continuous.
    To the right of t = 6, the graph is a straight line -- continuous.

    The only discontinuity would be at t = 6.
    To be a continuous function, the two curves must "meet" at t = 6.

    So we have: . f(4) \;=\;36-c \:=\:6c + 3\quad\Rightarrow\quad 7c \:=\:33

    Therefore: . c\,=\,\frac{33}{7}



    2.) For what value of c is the function f continuous?

    . . f(x) \;=\;\begin{Bmatrix}cx + 3, & x < 4 \\ cx^2-3, & x \geq 4\end{Bmatrix}

    The two branches of the function must be equal at x = 4.

    Hence: . 4c + 3 \:=\:16c - 3\quad\Rightarrow\quad12c \:=\:6

    Therefore: . c\,=\,\frac{1}{2}

    Last edited by Soroban; September 11th 2007 at 10:49 PM.
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  3. #3
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    Thanks for the great explanation. Clarified everything. Thanks again.
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