Dealing with absolute value functions

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SUMMARY

The discussion focuses on simplifying the expression involving absolute value functions, specifically f(x) = |x - 1 - 1| and g(x) = x^2 + 2x. The user seeks clarity on whether they can simplify g(x) - f(x) = (|x - 1| - 1) - (x^2 + 2x) further. They propose a method of solving the problem by considering both positive and negative cases of the modulus, which is a valid approach for handling absolute values in integrals.

PREREQUISITES
  • Understanding of absolute value functions
  • Knowledge of polynomial functions
  • Familiarity with integral calculus
  • Experience with piecewise functions
NEXT STEPS
  • Study the properties of absolute value functions in calculus
  • Learn about piecewise function definitions and their applications
  • Explore methods for integrating functions involving absolute values
  • Review techniques for simplifying expressions with modulus
USEFUL FOR

Students and educators in mathematics, particularly those studying calculus and algebra, as well as anyone working with functions involving absolute values.

sunfleck
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In order to get an integral I need to find the difference between two functions, but I'm not sure how to deal with the absolute value...


f(x) = \left|x-1-1\right|
g(x) = x^2 + 2x

g(x) - f(x) = (\left|x-1\right|-1) - (x^2 + 2x)
=...
I don't know if I can simplify it anymore... can I take that |x| out so - 2x + |x| = - x? If so what happens to the |-1| I feel like I probably can't simplify any further, but I'd like to know for sure
 
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The easiest way I find to solve problems like this is to solve the problem twice - once where the contents of the modulus are positive anyway, and once when they're negative (in which case you need to put an extra minus sign in front of them).
 

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