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how do i find the domain of f(x)=|x+1|?

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Greetings Alexis!

Any time you encounter absolute values, that is short hand for saying you have a piece-wise function, or two functions put together. In essence, the function you've been given is shorthand for:

f(x) = -(x+1) = -x-1 for x+1<0 --> x<-1


f(x) = +(x+1) = x+1 for x+1>=0 --> x>=-1

If that all seems too cluttered, just notice that all I am doing is taking the function inside the absolute value, and multiplying it by -1 for any value of x that makes it negative. I am leaving it alone otherwise.

Now you can see that |x+1| is a "downhill" line when x is less than -1, and an "uphill" line when x is greater than or equal to -1. No matter what the case, however, |x+1| is defined for all values of x, therefore the domain is all real numbers.

I know I went into some gory details there, but hopefully it has made absolute values a little less mysterious. They are quite possibly one of the strangest concepts to grasp in algebra.