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Math Help - Finding the constants a and b to make continuous on entire real line

  1. #1
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    Finding the constants a and b to make continuous on entire real line

    Given:

    f(x)=\left\{\begin{array}{cc}2,&\mbox{ if }<br />
x\leq -1\\ ax+b,& \mbox{ if } \--1 < x < 3\\ -2, & \mbox{ if } x\geq 3\end{array}\right.

    I need to find the constants a and b such that the function is continous on the entire real number line. how can i do this ? (step by step is helpful)
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  2. #2
    Senior Member pankaj's Avatar
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    Quote Originally Posted by hemi View Post
    Given:

    f(x)=\left\{\begin{array}{cc}2,&\mbox{ if }<br />
x\leq -1\\ ax+b,& \mbox{ if } \--1 < x < 3\\ -2, & \mbox{ if } x\geq 3\end{array}\right.

    I need to find the constants a and b such that the function is continous on the entire real number line. how can i do this ? (step by step is helpful)
    A function is said to be continuous at x=a if \lim_{h\to 0}f(a-h)=\lim_{h\to 0}f(a+h)=f(a)

    \lim_{h\to 0}f(-1-h)=-2

    \lim_{h\to 0}f(-1+h)=\lim_{h\to 0}(a(-1+h)+b)=b-a

    Therefore, b-a=-2



    \lim_{h\to 0}f(3-h)=\lim_{h\to 0}(a(3-h)+b)=3a+b

    \lim_{h\to 0}f(3+h)=\lim_{h\to 0}(-2)=-2

     <br />
3a+b=-2<br />

    Solving for a and b we get a=0 and b=-2
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  3. #3
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by pankaj View Post
    A function is said to be continuous at x=a if \lim_{h\to 0}f(a-h)=\lim_{h\to 0}f(a+h)=f(a)

    \lim_{h\to 0}f(-1-h)=""-2""

    \lim_{h\to 0}f(-1+h)=\lim_{h\to 0}(a(-1+h)+b)=b-a

    Therefore, b-a=""-""2



    \lim_{h\to 0}f(3-h)=\lim_{h\to 0}(a(3-h)+b)=3a+b

    \lim_{h\to 0}f(3+h)=\lim_{h\to 0}(-2)=-2

     <br />
3a+b=-2<br />

    Solving for a and b we get a=0 and b=-2
    just a small thing pankaj b-a=2 not -2 since f(-1)=2
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  4. #4
    Senior Member pankaj's Avatar
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    Sorry. I goofed up
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  5. #5
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    will this make the final answer of b = 2 then rather then -2?
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  6. #6
    MHF Contributor Amer's Avatar
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    b-a=2
    3a+b=-2 ok

    a-b=-2
    3a+b=-2 find sum two equation

    4a=-4
    a=-1 so

    b-(-1)=2
    b+1=2
    b=1
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