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Math Help - Make the function continuous everywhere

  1. #1
    Senior Member vaironxxrd's Avatar
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    Make the function continuous everywhere

    I'm sorry to come back with another Calculus question, to those of you who see me asking often.

    I'm trying to make the following function continuous.

     f(x) = -1 if  x <= 0


    ax+b if 0 < x < 1


    1 if x >= 1

    I've seen simple examples which use limits to find the values but here I believe I would have to use the following limits, which will produce different values.

    \lim_{x \to -1} ax+b = -1

    \lim_{x \to 1} ax+b = 1

    I honestly don't think this makes any sense.
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  2. #2
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    Re: Make the function continuous everywhere

    Quote Originally Posted by vaironxxrd View Post
    I'm trying to make the following function continuous.
     f(x) = -1 if  x <= 0
    ax+b if 0 < x < 1
    1 if x >= 1
    You need {\lim _{x \to {0^ + }}}ax + b =  - 1 and {\lim _{x \to {1^ - }}}ax + b =  1
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    Senior Member vaironxxrd's Avatar
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    Re: Make the function continuous everywhere

    Quote Originally Posted by Plato View Post
    You need {\lim _{x \to {0^ + }}}ax + b =  - 1 and {\lim _{x \to {1^ - }}}ax + b =  1
    But that means we would get two answers?

    \lim_{x \to 0^+} ax+b = -1

    = a(0)+b = -1

    = b = -1

    and

    \lim_{x \to 1^-}ax+b=1

    = \lim_{x \to 1^-}a(1)+b=1

    = \lim_{x \to 1^-}a+b=1

    I think I have to review some algebra 1 material
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    Re: Make the function continuous everywhere

    Quote Originally Posted by vaironxxrd View Post
    But that means we would get two answers?

    \lim_{x \to 0^+} ax+b = -1

    = a(0)+b = -1

    = b = -1

    and

    \lim_{x \to 1^-}ax+b=1
    Look you know that b = -1

    so find the a that makes \lim_{x \to 1^-}ax-1=1
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    Senior Member vaironxxrd's Avatar
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    Re: Make the function continuous everywhere

    Quote Originally Posted by Plato View Post
    Look you know that b = -1

    so find the a that makes \lim_{x \to 1^-}ax-1=1
    So to make this function continuous from 0<x<1...

    b = -1
    \lim_{x \to 1-} ax-1 = 1

    = a(1)-1 = 1

    = a -1 = 1

    = a = 2

    Therefore from x<0<1 the function is defined as ax+b = 2x -1
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    Re: Make the function continuous everywhere

    Quote Originally Posted by vaironxxrd View Post
    So to make this function continuous from 0<x<1...

    b = -1
    \lim_{x \to 1-} ax-1 = 1

    = a(1)-1 = 1

    = a -1 = 1

    = a = 2

    Therefore from x<0<1 the function is defined as ax+b = 2x -1
    f(x)=\left\{ {\begin{array}{rc}{ - 1,}&{x < 0}\\{2x - 1,}&{0 \le x \le 1}\\{1,}&{1 < x}\end{array}} \right.
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