Find a and b for Continuous Function on Real Line

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    Continuous Function
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Discussion Overview

The discussion revolves around determining the constants a and b for a piecewise function to ensure its continuity across the entire real line. Participants explore the implications of their proposed values and the conditions for continuity at specific points.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests a = 1 and b = -1 as potential solutions for the constants.
  • Another participant questions the correctness of the proposed values, prompting a re-evaluation.
  • A participant highlights the need to check the limits of the function at the boundaries x = -1 and x = 3 to ensure continuity.
  • One participant presents a system of linear equations derived from the continuity conditions: -a + b = 2 and 3a + b = -2.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the values of a and b, and multiple competing views remain regarding their correctness and the conditions for continuity.

Contextual Notes

Participants express uncertainty about the correctness of the proposed values and the implications of continuity at specific points, indicating that further verification is needed.

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Q.

Determine the constants a and b so that the function is continuous on the entire real line.

2 if x <= -1
f(x) = ax+b if -1 < x < 3
-2 if x >= 3

Ans:

a = 1; b = -1

I wonder if the answer is right??
 
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fr33pl4gu3 said:
a = 1; b = -1

I wonder if the answer is right??

Almost … check it again … :wink:
 
Which one is wrong, a or b, or both??
 
Suppose your answer IS correct, that would mean

2 if x <= -1
f(x) = x - 1 if -1 < x < 3
-2 if x >= 3

However when x = -1, you get 2 (from the criterion x <= -1) and -2 for the criterion -1 < x < 3. Also when x = 3, you get 2 for the criterion -1 < x < 3 and -2 for the criterion x >= -3. You don't want those jumps if your f is cont.
 
Exactly. You shouldn't have to "wonder" if your answer is correct, you can check it yourself. Is the left-side limit of f(x) equal to the right-side limit at x=-1? at x = 3?
 
you have 2 linear equations with 2 unknowns

-a + b = 2
3a + b = -2
 

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