# How to write absolute value expressions as piecewise expressions

If it was less than or equal, then the function would have been defined at x equals -9, but it's not. Jo Steig The absolute value symbol is really shorthand for what we call a piecewise described function.

Write 2x - 3 without using absolute value The following three examples illustrate 1. The end result will be a piecewise defined function that is similar to the original definition that was given above for absolute value.

I always find these piecewise functions a lot of fun. Therefore, to get any work done we must first write it without absolute values.

It's very important to look at this says, -9 is less than x, not less than or equal. So it's very important that when you input - 5 in here, you know which of these intervals you are in. This graph, you can see that the function is constant over this interval, 4x.

### How to write absolute value expressions as piecewise expressions

So let me give myself some space for the three different intervals. Our decompression program for the absolute value is the definition given above. Not a closed in circle. At this point we are just trying to develop the skill of writing an absolute value expression without the absolute value symbol. You will investigate addition, integration, and differentiation in a later course. Sometimes people call this a step function, it steps up. We have an open circle right over there. It jumps up here.

That's this interval, and what is the value of the function over this interval? We just need to insure that out output is nonnegative. At this point we are just trying to develop the skill of writing an absolute value expression without the absolute value symbol.

Before you can access the information you must first run a decompression program. Now let's keep going.

### Graphing absolute value piecewise functions

Let's take a look at this graph right over here. We have this last interval where we're going from -1 to 9. So not including -9 but x being greater than -9 and all the way up to and including When the inside expression includes variables and you do not know anything about the values of the variables, then writing the expression without absolute values takes a bit more care and our reliance on the definition above becomes more obvious. It's very important to look at this says, -9 is less than x, not less than or equal. The end result will be a piecewise defined function that is similar to the original definition that was given above for absolute value. A constant -7 and we're done. And x starts off with -1 less than x, because you have an open circle right over here and that's good because X equals -1 is defined up here, all the way to x is less than or equal to 9. In the original definition for absolute value, replace each occurance of 'a' with 4 - 5x. You can't be in two of these intervals. Not a closed in circle. It's only defined over here. The following three examples illustrate 1. Actually, when you see this type of function notation, it becomes a lot clearer why function notation is useful even. But what we're now going to explore is functions that are defined piece by piece over different intervals and functions like this you'll sometimes view them as a piecewise, or these types of function definitions they might be called a piecewise function definition.

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