Dealing with absolute value functions

AI Thread Summary
To find the integral of the difference between the functions f(x) = |x-2| and g(x) = x^2 + 2x, it's essential to address the absolute value in f(x). The discussion suggests simplifying g(x) - f(x) but raises concerns about handling the absolute value correctly. A recommended approach is to evaluate the problem in two scenarios: when the expression inside the absolute value is positive and when it is negative, applying a negative sign as needed. This method ensures a comprehensive understanding of the function's behavior across different intervals. Properly managing the absolute value is crucial for accurate integration.
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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