Discuss continuity of the composite function

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Homework Help Overview

The discussion revolves around the continuity of composite functions, specifically examining the functions h(x) = f(g(x)) with given definitions for f(x) and g(x). Participants are exploring the conditions under which these functions remain continuous.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning how to determine the point 'a' for continuity without a specific value provided. There is a focus on identifying where the function f(x) = 1/(x-6) is discontinuous and discussing the implications of different interpretations of the function's definition.

Discussion Status

The discussion is active, with participants providing insights into the continuity of the functions involved. Some have suggested examining all possible values for continuity, while others are clarifying misunderstandings regarding specific points of discontinuity. There is no explicit consensus yet, but productive questions are being raised.

Contextual Notes

Participants are navigating potential ambiguities in the definitions of the functions and the implications of continuity at various points, particularly around x = 6 and x = -6. The lack of a specific point 'a' adds complexity to the discussion.

louie3006
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Homework Statement

:
discuss continuity of the composite function h(x)=f(g(x)) when A} F(x)=X^2 , g(x) = x-1
B} f(x) = 1/x-6 , g(x) = X^2+5
where should I start ?
 
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h is continuous at a point a if f is continuous at g(a)
 


and how can we find out what a is if we are not given a number for a ?
 


Consider all numbers! At what values of x is f(x)= 1/(x- 6) NOT continuous? (That's the crucial question.)
 


when x = -6, that would make f(x) = 1/x-6 discontinuous right?
 


louie3006 said:
when x = -6, that would make f(x) = 1/x-6 discontinuous right?
Please use parentheses.
If you mean f(x)= (1/x)- 6, then f(-6)= -1/6- 6= -37/6. No this function is not discontinuous at -6.

If you mean f(x)= 1/(x- 6), then f(-6)= 1/(-6-6)= -1/12. No this function is not discontinuous at -6.

Be more careful!
 


f(6) = 1/ (6-6) = 1/0 this function is discontinuous, right ?
 

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