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## Piecewise functions

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# Worked example: domain & range of piecewise linear functions

CCSS Math: HSF.IF.A.1

## Video transcript

So we have a piecewise linear function
right over here for different intervals of x. g of x is defined by a a line or the line changes depending
what interval of x we're actually in. And so let's think about its domain, and
then we'll think about its range. So the domain of this, this is a review.
The domain is the set of all inputs for which this function is defined, and our input
variable here is x. This is a set of all x values for which
this function is defined. And we see here. Anything, anything negative 6 or
lower, our function isn't defined. If it, if x is negative 6 or or lower than that. I don't --
it doesn't, it doesn't fall into one of these three intervals. So there is no definition for it.
It doesn't say hey do this in all other cases for x. It is just saying, look, if x falls into one of these three
conditions, apply this. And if x doesn't fall into one of those three
conditions, well this function g is just not defined. So, to fall into one of these three, you have
to be at least greater than negative 6. So this part right over here, the low end
of our domain is defined right over there, so we say, we could say, negative 6 is less than x and I'm leaving --
so let's write it here. All real numbers -- actually
let me write this way x, I could write it more math-y. I could say x is a member of the real numbers such that, such that negative 6 is less than x. Negative 6 is less than x and I also think about the upper bound.
So as x goes, I just wanna make sure that we fill in all the gaps between x being a greater than negative 6 and
x is less than or equal to 6. So let's see. As we go up to and including negative 3,
we're in this clause. As soon as we cross negative 3, we fall into this clause up to 4, but
as soon as we get 4, we're in this clause up to and including 6. So x at the high end is said to be less than
or equal to 6, less then or equal to 6. Now another way to say this and kind of less math-y notation is x, x can be any real number, any the real number such that, such that negative 6 is less than x is less than or equal to 6. These two
statements are equivalent. So now let's think about
the range of this function. Let's think about the range, and the range
is, this is the set of all inputs , oh sorry, this is the set of all
outputs that this function can take on, or all the
values that this function can take on. And to do that, let's just think about
as x goes, but x varies or x can be any values in this
interval. What are the different values
that g of x could take on? Let's think about that. g of x is going
to be between what and what? g of x is going to be between what and
what? g of x is going to be between what and
what? And it might actually, this might be some
equal signs there but I'm gonna worry about that in a second. So when does this thing hit its low
point? o this thing hits, hits its low point when x is as small as possible. An x is
going to be as small as possible when x is approaching negative 6. So if x were equal to negative 6, it can't
equal negative 6 herer but if x is equal to negative 6, then this thing over here
would be equal to negative 6 plus 7, would be, would be 1. So if x is greater than negative 6, g of x is going to be greater than 1, or another way
to think about it is if negative 6 is less than x, then 1 is going to be less than g of x. And the reason I said that is if I put negative 6
into this, negative 6 plus 7 is equal to 1. Now this gonna hit a
high end when it as large as possible. The largest value in this interval that
we can take on is x being equal to negative 3.
So when x is equal to negative 3, negative 3 plus 7 is equal to 4, positive 4. And it can actually take on
that value because this is less than or equal to, so we can actually take on
x equals negative 3 in which case g of x actually
will take on positive 4. So, let' do that for each of these.
Now here we have 1 minus x, so this is going to take on
its smallest value when x is as large as possible. So the largest value x can approach for,
it can't quite take on for, but it's going to approach for. So if x, let's see, if we said x was 4,
although that's not this clause here, 1 minus x, 1 minus 4 is negative 3. So as long as x is less than 4, then negative 3 is going to be less than g of x. I wanna make sure that makes sense
to you because it can be little bit confusing because this takes on its minimum value when x is approaching, or it's
approaching its minimum value when x is approaching its, when x is
approaching its maximum because we're subtracting it. So if you take the upper end, even though
this doesn't actually include 4, but as we approach for, we could say, OK, 1 negative 4 is negative 3 so that's, so g of x
is always going to be greater than that, as well it's going to be
going to be a less than. Well what happens as we approach x being equal to negative 3? So, 1 minus negative 3 is going to be positive 4. So this is going to be positive
4 right over here. And these are both less than, not less
than or equal to because these are both less than right
over here. And now let's think about this right over here. So 2x minus 11 is gonna hit its maximum value
when x is as large as possible. So its maximum value's going to be hit
when x is equal to 6 So 2 times 6 is 12, minus 11.
Well that's going to be 1. So its maximum value's going to be 1.
It's actually going to be able to hit because x can be equal to 6. Its minimum value is going to be when x
is equal to 4, and actually can be equal to 4. We have this less than or equal sign right over there.
So 2 times 4 is 8, minus 11 is negative 3. So, g of x in this case
can get as low as negative 3 when x is equal to 4. So now let's think about all of, all of the
values that g of x can take on. So we could say, we could write this a bunch of ways,
we could write g of x is going to be a member of the real
numbers such that -- let's see. What's the lowest
value g of x can take on? g of x can get as low as negative 3.
It can even be equal to negative 3. This one just has been greater the
negative 3, but here can be greater than or equal to negative 3.
So negative 3 is less than or equal to g of x, and it
can get as high as, it can get as high as Let's see. It's defined all the way to
1 and then -- or I shouldn't say it is defined all the way to 1.
It can take on values up to 1 but it can also take on values beyond 1.
It can take on values all the way up to including 4 over here. So it can take on values up to and including 4. So g of x is a member
real numbers such that negative 3 is less than or equal to g of x
is less than or equal to 4. So the set of all values
that g of x can take on between, including and including negative 3 and positive 4.