MHB How do you define absolute value function on different intervals?

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The absolute value function f(x) = |x+3| - |x-3| is defined piecewise across three intervals: (-∞, -3), [-3, 3), and [3, +∞). For the interval (-∞, -3), the function simplifies to f(x) = -6. The discussion seeks to define f(x) for the other two intervals, specifically asking for guidance on how to approach this piecewise definition. Participants are encouraged to apply the properties of absolute values to determine the function's behavior in these remaining intervals. The focus remains on accurately defining the function across all specified ranges.
paulmdrdo1
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Define f(x)= |x+3|-|x-3| without absolute value bars piecewise in the following intervals (-∞,-3);[-3,3);[3,+∞).

this is how i do the problem,

I removed the absolute value bars first

f(x)= x+3-x+3 = 6

now i don't know how to define it piecewise. can you show me how define it correctly. thanks!
 
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Re: absolute value function.

On the interval:

i) $$(-\infty,-3)$$

we have:

$$x+3<0\,\therefore\,|x+3|=-(x+3)$$

$$x-3<0\,\therefore\,|x-3|=-(x-3)$$

and so $$f(x)=(-(x+3))-(-(x-3))=-6$$

Can you try the other two intervals?
 
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