# Domain and range of a function given a formula | Algebra II | Khan Academy

Determine the domain and range of the function f of x is equal to

3x squared plus 6x minus 2. So, the domain of the function is:

what is a set of all of the valid inputs, or all of the valid x values

for this function? And, I can take any real number,

square it, multiply it by 3, then add 6 times that real number

and then subtract 2 from it. So essentially any number if we’re talking

about reals when we talk about any number. So, the domain, the set

of valid inputs, the set of inputs over which this function

is defined, is all real numbers. So, the domain here is

all real numbers. And, for those of you who might say, well,

you know, aren’t all numbers real? You may or may not know that

there is a class of numbers, that are a little bit bizarre

when you first learn them, called imaginary numbers

and complex numbers. But, I won’t go into that right now. But, most of the traditional

numbers that you know of, they are part of

the set of real numbers. It’s pretty much

everything but complex numbers. So, you take any real number

and you put it here, you can square it, multiply it by 3,

then add 6 times it and subtract 2. Now, the range, at least the way we’ve

been thinking about it in this series of videos– The range is set of possible,

outputs of this function. Or if we said y equals f of x

on a graph, it’s a set of all the possible y values. And, to get a flavor for this,

I’m going to try to graph this function right over here. And, if you’re familiar with quadratics– and that’s what this function is

right over here, it is a quadratic– you might already know

that it has a parabolic shape. And, so its shape might look

something like this. And, actually this one will

look like this, it’s upward opening. But other parabolas

have shapes like that. And, you see when a parabola

has a shape like this, it won’t take on any values

below its vertex when it’s upward opening, and it won’t take on any values above

its vertex when it is downward opening. So, let’s see if we can graph

this and maybe get a sense of its vertex. There are ways

to calculate the vertex exactly, but let’s see how we can

think about this problem. So, I’m gonna try some x and y values. There’s other ways to directly compute the

vertex. Negative b over 2a is the formula for it. It comes straight out of the quadratic formula, which you get from completing the

square. Lets try some x values and lets see what f

of x is equal to. So, let’s try, well this the values we’ve

been trying the last two videos. What happens when x is equal to negative

two? Then f of x is 3 times negative 2 squared,

which is 4, plus 6 times negative 2, which is 6 times negative 2,

so it’s minus 12 minus 2. So, this is 12 minus 12 minus 2. So, it’s equal to negative 2. Now, what happens when x is equal to

negative 1? So, this is going to be 3 times negative 1

squared, which is just 1, minus, or I should say plus 6 times negative 1 which

is minus 6 and then minus 2, and then minus

2. So, this is 3 minus 6 is negative 3 minus 2 is equal negative 5, and that actually

is the vertex. And, you know the formula for the vertex,

once again, is negative b over 2 a. So, negative b. That’s the coefficient on this term right

over here. It’s negative 6 over 2 times this one

right over here, 2 times 3. 2 times 3, this is equal to negative 1. So, that is the vertex, but let’s just

keep on going right over here. So, what happens when x is equal to 0? These first two terms are 0, you’re just

left with a negative 2. When x is equal to positive 1. And, this is where you can see that this is the vertex, and you start seeing the

symmetry. If you go one above the vertex, f of x is

equal to negative 2. If you go one x value below the vertex, or

below the x value of the vertex, f of x is equal to

negative 2 again. But, let’s just keep going. We could try, let’s do one more point over

here. So, we have, we could try, x is equal to

1. When x is equal to 1, you have 3 times one

squared which is 1. So, 3 times 1 plus 6 times 1, which is

just 6, minus 2. So, this is 9 minus 2 it’s equal to 7. And, that I think is enough points to give us a scaffold of what this graph will look

like. What the graph of the function would look

like. So, it would look something like this. I do my best to draw it. So, this is a x equals negative 2. We draw the whole axis. This is x is equal to negative 1, this is

x is equal to, this is x is equal to 0 and then this is x is equal to

1 right over there and then when x is equal to, we go from negative 2

all the way to positive. Or, we should go from negative 5 all the

way to positive 7. So, let’s say this is negative 1,2,3,4,5. That’s negative five over there on the y

axis, y axis and then it will go to positive 7. One, two, three, four, five, six, seven. I could keep going, this is in the y, and

we’re going to set y equal to whatever our output of

the function is. Y is equal to f of x. And this is one right here. So, lets plot the points. You have the point negative 2, negative 2. When x is negative 2, this is the x axis. When x is negative 2, y is negative 2. Y is negative 2 so that is that right over

3. So, that is the point, that is the point

negative 2, negative 2. Fair enough? Then, we have this point that we have this

pink or purplish color. Negative, when x is negative 1, f of x is

negative 5. When x is negative 1, f of x is negative

5. And, we already said that this is the

vertex. And, you’ll see the symmetry around it in

a second. So, this is the point negative 1, negative

5. And then, with the point 0, negative 2. 0, negative when x is a 0, y is negative

2, for f’ of x is negative 2 or f of 0 is negative 2, so this is the

point 0, negative 2, and then finally when x is equal to 1 and

f of 1 is 7, f of 1 is 7. So, that’s right there it’s a point 1, 7

and it gives us a scaffold for what this parabola, what this

curve will look like. So, I’ll try my best to draw it

respectably. So, it would look something, something

like that, and keep on going in that direction. Keep on going in that direction. But, I think you see the symmetry around

the vertex. That if you were to. If you were to put a line right over here,

the two sides are kind of the mirror images of

each other. There, you can flip them over, and that’s

how we know it’s the vertex. And, that’s how we also know, because this

is an upward opening parabola, I mean, there is formulas for vertex, and

there are multiple ways of calculating it. But, since it’s an upward opening

parabola, where the vertex is going to be, the minimum

point. This is the minimum value that the

parabola will take on. So, going back to the original question,

this is all for trying to figure out the range, the set of y values, the set of

outputs that this function can generate. You see that the function, it can get as

low as negative 5. It got all the way down to negative 5

right at the vertex. But, as you go to the right, as x values

increase to the right or decrease to the left, then

the parabola goes upwards. So, the parabola can never give you

values– f of x is never going to be less than

negative 5. So, our domain,

but it can take on all the vaues. It can keep on increasing forever as x

gets larger, x gets smaller farther away from

the vertex. So, our range, so we already said our

domain is all real numbers. Our range, the possible y values

is all real numbers greater than or equal to negative

5. It can take on the value of any real number greater than or equal to

negative 5. Nothing less than negative 5.

Why did u start at -2 when determining the domain and range

we had this during grade 8 but I totally forgot all about it and because school just started we are reviewing past lessons. my teacher makes it really confusing and his grammar makes it a whole lot worse! thank you!! you're a life saver

this is very helpful, thankyou!

So the domains are the inputs????? Please answer

u sound like dj vlad

Sal,your voice,makes me happy😃😃

its actually +2 in the first case as

-2_(-1)= 2

How did you get all real numbers for the domain? I always get confused on that.

Thank yooou! i have an exam in 3 days and this really helped!

when obama can teach you math

Illuminati Confirmed!😁😁😡😱✡🔼

I understand why the range is all real numbers, but why is it all real numbers greater than or equal to -5

I have so much respect for you wondering how you are able to retain so much information

For the range, is it always bound to the y axis? and domain in the x axis? Or am I just imagining false information that will mislead me?

this comment section is cancer

you can write the range like [-5,+inf) and thank you

WHY THE FUCK IS THERE NOTHING ON THIS STUPID WEB THAT EXPLAINS FUNCTIONS WITH 2 VARIABLES. E.g. f(4) – e(.5) with f(x)=3x+1 & e(x)=2 over x. Anyone know? MY EXAM IS TOMORROW AND I WAS NEVER TAUGHT THAT YET ITS ON MY REVISION SHEET

how can you tell if the parabola is upward or downward just by looking at the quadratic expression?

my teacher so dumb He makes me hate math rn.

very very helpful thankyou sir 😀

I cannot thank you enough for these videos. They are extremely helpful to me. I need all the help I can get 😀

Make videos in hindi

Who here is in middle school?

from the bottom of my heart, thank you for helping me throughout the years

what about find the range of 4/2x+1

Wow! Sir you are great

Thank you! I have an exam tomorrow.

thank you

Studying for a math test this helped…

Best explanation! Finally understood!!

Mental

Abuse

To

Humans

This problem is sooo easy.

Try to do some more harder problems is appreciable.

Like the below ones_

_F(X)= 1

————–

√[(|x-4|)-3]

And log functions

Modulus functions.

Try to something different yar no one asks such easy questions in exam.

This is 4G and 1G

there are infinitely faster and easier ways to explain how to do this, waste of time

He talks too fast for me

Thank you thank you this is easy 🙏💕

this is very good for finding domain and range guys i recommend it to students that need more help

Something…something like this

…Tododo tododo…

Thanx a-lot u are helping me .. thank God 🙏for having good ppl who would love ❤️ to help others for nothing .

OMG This is actually so good

Hi Sal, pretty well with the graph explanation but make it a bit simpler. Find f(x)=y such that the determinant is negative and the number turns imaginary. Since for y=-5, you get 3(x-1)^2 and hence the determinant must be 0 because there are equal roots, any y below -5 would make the discriminant 0 and hence the range is [-5,∞). This is what I was taught in 8th grade.

im just stupid I think, ive watched so many videos and I still don't understand

Lost me at determine

Oh thank God for your professionalism…. Jesus christ there are so. Many IDIOTS!

I love this man

In my hw, the domain is x>=0. What should i do?

Thanks man 😊😊😊

what about the domain? the x value keeps going up forever?

All imagery numbers are real because if you stretch a parabola into 3 dimensional space then imagery numbers appear therefor they are real

how can input be negative we are putting negtive 2 as input is it defined

i have khan academy due at 5 am ;( pls help me ill pay u if u do it for me

wow..thanks you..nice explaination

Ive got a math test tomorrow thx so much for thd help

What is the range of y=x4 (exponent "4") + x2 (exponent "2") , please help me

U need to teach -d/4a to find vertex that's what I was here for

I didn't understand anything

Please explain properly

my brain is gonna explode

Where did you get your -2,-1,0,1?

thanks. our math teacher sucks

Wow I learned this lesson in just 8 minutes while my prof was teaching for 1 hour I can't even understand a single thing. Thanks man you're the real MVP!

What if it's only f(x)=x+1???

What if your function is a cubic one , how can you get the range of it?

Thank You so much. I am in Alg 1 and I am learning Alg 2. Your video was very helpfull.

Thanks alot

Thx Obama

He always makes it a lot harder than it supposed to be.

What happens if it says another variable inside the parenthesis do i replace it with x or…

This is so helpful omg😂🙏❤️

What would you do if you failed in the same class twice?

brilliant lesson!!!

How do you know if the domain is all real numbers?

3:34 it should be -3-6-2

Math is hard…

Thanks .. 💝

Wut???????????

in collage alg. wheres the interval notation version

You speak English but you weren't speaking English.

Who else in 7th grade??

Why does it feel like I'm always watching math videos but never actually retaining the information?

I wish he provided more examples. This function is all real numbers because it doesn't contain neither a square root or a division. simple.

I understand domain, but range is still confusing. Like why can’t it be lower than -5?

Thanks a lot for your videos

My A level exam for Pure Mathematics 1 starts in less than 5 hours so this helped

What is a Pre-Calculus Honors (Online) Student doing in a heck of a place like this?

i use this technique to get the range of a quadrantic function: if a > 0 then its y is greater than or equal to 4ac-b^2 all over 4a, if a < 0 then y is less than or equal to 4ac-b^2 all over 4a.

substitute the given function. 3x^2 + 6x – 2, a=3, b=6, c= – 2

4(3)(-2)-(6*6) = – 60 over (4)(3) = – 5, since in the given a>0, the range of the function is {all real numbers wherein y is greater than or equal to -5}

its easier to use the formula

how do you know it's the vertex?

What.

our Domain and range only has {x|x} or {y|y}, no graphs or table of values, how about that???

I don't really know if my brain hates hard math or what. This is really hard bruh i don't like this. 😭

Thank you so much since I just started in the K to 12 curriculum, I have prelims to answer this for I'm not a good mathematician but a great listener. Thx

2019 squad where you at

Thxx khan

any shortcuts for range? i don't have time for a graph when i have to do like 15 questions in 30 minutes

I think that only my teacher knows the trick that she showed me it takes fewer times to do but more difficult to understand

You shoudn't put the arrow on the curve

good

Am I the only one who didn’t understand it lol

I’m watching this through my tears lolThanks

His writing is so tiny, neat, and cute.