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.

## 15 Comments

## that_pac12

Congratulations on the new free SAT course!!!

## Le Quasar

excellent. thanks

## Christopher Caspersen

I think he changed the "such as" model to "|" from ":"

🙂

## anh le

5th comment 🙂

## Kasha Lackey

I'm still confused about the range part

## Bidoof

Thanks helped a lot

## Mack Sauce

BRO YOU ARE BLOWING MY MIND!

## mathSci

Hey ! how a function can have two different value at a point.? -3 to 1 is in two piecewise function. ???? Please help

## Mohamed Nasser

thanks

## Rosé Manoban

What if it says 1 when x<_0?

## Yong QH

Is this works for quadratic functions?

## Austin Chandler

Ya know, a graph is much easier.

## Pem aken

Too clutch

## Begimai Temiralieva

"I can write it more mathy", idk why I think it was cute

## Vina Sanjoyo

Thank you, this is so helpful!