Extension 2.7 writing and graphing piecewise functions
And now let's look at this last interval.
And that point, and it includes, so x is defined there, it's less than or equal to, and then we go all the way to negative two. Actually, when you see this type of function notation, it becomes a lot clearer why function notation is useful even.
When x is negative one, we are approaching, or as x approaches negative one, we're approaching negative one plus seven is six.
And so this one actually doesn't have any jumps in it. This is negative 66 over 11, is that right?
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Piecewise functions Video transcript - [Voiceover] So, I have this somewhat hairy function definition here, and I want to see if we can graph it. This graph, you can see that the function is constant over this interval, 4x. It looks like stairs to some degree. And then I'm gonna draw the line. So it's very important that when you input - 5 in here, you know which of these intervals you are in. The next interval, this one's a lot more straightforward. The next interval is from -5 is less than x, which is less than or equal to And this is a piecewise function. It's a constant -9 over that interval. I always find these piecewise functions a lot of fun. So let me give myself some space for the three different intervals. It's only defined over here.
It's up to and including, it's up to negative two, not including. 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.
We have this last interval where we're going from -1 to 9. We have graphed this function that has been defined in a piecewise way. I just multiplied this times 10, 12 times 10 isand we have the negative, plus 54 over It's only defined over here.
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And that point, and it includes, so x is defined there, it's less than or equal to, and then we go all the way to negative two. This graph, you can see that the function is constant over this interval, 4x. We have just constructed a piece by piece definition of this function. 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. I encourage you, especially if you have some graph paper, to see if you can graph this on your own first before I work through it. So let me give myself some space for the three different intervals. So, let's think about this first interval. Sometimes people call this a step function, it steps up. So, I'm gonna put a little open circle there, and then I'm gonna draw the line. Now let's keep going. Not a closed in circle. It's very important to look at this says, -9 is less than x, not less than or equal. That's this interval, and what is the value of the function over this interval? Now, we might be tempted, we might be tempted, to just circle in this dot over here, but remember, this interval does not include negative two. Now this first interval is from, not including -9, and I have this open circle here.
So, we're able to fill in that right over there, and then when x is equal to 10, you have negative over So that's why it's important that this isn't a -5 is less than or equal to.
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