– As a little bit of

a review, we know that if we have some function,

let’s call it “f”. We don’t have to call it

“f”, but “f” is the letter most typically used for functions, that if I give it an input, a valid input, if I give it a valid input,

and I use the variable “x” for that valid input, it is

going to map that to an output. It is going to map that, or produce, given this x, it’s going

to product an output that we would call “f(x).” And we’ve already talked a little bit about the notion of a domain. A domain is the set of all of the inputs over which the function is defined. So if this the domain here,

if this is the domain here, and I take a value here,

and I put that in for x, then the function is

going to output an f(x). If I take something that’s

outside of the domain, let me do that in a different color… If I take something that

is outside of the domain and try to input it into this function, the function will say, “hey, wait wait,” “I’m not defined for that thing” “that’s outside of the domain.” Now another interesting

thing to think about, and that’s actually what

the focus of this video is, okay, we know the set of

all of the valid inputs, that’s called the domain,

but what about all, the set of all of the

outputs that the function could actually produce? And we have a name for that. That is called the range of the function. So the range. The range, and the most typical, there’s actually a couple

of definitions for range, but the most typical definition for range is “the set of all possible outputs.” So you give me, you input

something from the domain, it’s going to output

something, and by definition, because we have outputted

it from this function, that thing is going to be in the range, and if we take the set

of all of the things that the function could output, that is going to make up the range. So this right over here is

the set of all possible, all possible outputs. All possible outputs. So let’s make that a

little bit more concrete, with an example. So let’s say that I have the function f(x) defined as, so once again,

I’m gonna input x’s, and I have my function f,

and I’m gonna output f(x). And let’s say this def… The function definition

here, the thing that tries to figure out, “okay, given an

x, what f(x) do I produce?”, the definition says “f(x)

is going to be equal” “to whatever my input is, squared.” Well, just as a little bit of review, we know what the domain

here is going to be. The domain is the set of all valid inputs. So what are the valid inputs here? Well, I could take any real number and input into this, and I

could take any real number and I can square it, there’s

nothing wrong with that, and so the domain is all real numbers. All, all real, all real numbers. But what’s the range? Maybe I’ll do that in a different color just to highlight it. What is going to be the range here, what is the set of all possible outputs? Well if you think about, actually, to help us think about, let

me actually draw a graph here. Of what this looks like. What this looks like. So the graph of “f(x)

is equal to x squared” is going to look something like this. So, it’s gonna look, it’s going

to look something like this. I’m obviously hand-drawing

it, so it’s not perfect. It’s gonna be a parabola with a, with a vertex right here at the origin. So this is the graph, this is the graph, “y is equal to f(x),” this

of course is the x-axis, this of course is the y-axis. So let’s think about it, what is the set of all possible outputs? Well in this case, the set

of all possible outputs is the set of all possible y’s here. Well, we see, y can take

on any non-negative value. y could be zero, y could one,

y could be pi, y could be e, but y cannot be negative. So the range here is, the range… We could, well we could

say it a couple of ways, we could say, “f(x)”,

let me write it this way. “f(x) is a member of the real numbers” “such that, is such that

f(x) is greater than” “or equal to zero.” We could write it that

way, if we wanted to write it in a less mathy notation, we could say that “f(x) is going to be” “greater than or equal to zero.” f(x) is not going to be negative, so any non-negative number, the set of all non-negative numbers, that is our range. Let’s do another example of this, just to make it a little bit,

just to make it a little bit, a little bit clearer. Let’s say that I had,

let’s say that I had g(x), let’s say I have g(x),

I’ll do this in white, let’s say it’s equal

to “x squared over x.” So we could try to

simplify g(x) a little bit, we could say, “look, if I have x squared” “and I divide it by x, that’s gonna,” “that’s the same thing as

g(x) being equal to x.” “x squared over x” is x,

but we have to be careful. Because right over here, we have to, in our domain, x cannot be equal to zero. If x is equal to zero,

we get zero over zero, we get indeterminate form. So in order for this function

to be the exact same function, we have to put that,

’cause it’s not obvious now from the definition, we have to say, “x cannot be equal to zero.” So g(x) is equal to x for any x as long as x is not equal to zero. Now these two function

definitions are equivalent. And we could even graph it. We could graph it, it’s going to look, I’m gonna do a quick and

dirty version of this graph. It’s gonna look something like, this. It’s gonna have a slope of one, but it’s gonna have a hole right at zero, ’cause it’s not defined at zero. So it’s gonna look like this. So the domain here, the domain of g is going to be, “x is a

member of the real numbers” “such that x does not equal zero,” and the range is actually

going to be the same thing. The range here is going to be, we could say “f(x) is a

member of the real numbers” “such that f(x) does not equal zero.” “f(x) does not equal zero.” So the domain is all real

numbers except for zero, the range is all real

numbers except for zero. So the big takeaway here is

the range is all the pos… The set of all possible

outputs of your function. The domain is the set of all valid inputs into your function.

This video MUST NOT BE INDEXED in readily accessible places . I tried finding it on khanacademy.org putting the exact title "Introduction to range of a function" with and without quotes into the Search Box. I could not find it in the first 10 pages of videos (100 videos). Google Site Search did not turn up anything. Googling on the web it with quotes only turned up this exact same YouTube video. So, I can watch the video – get educated – which I am grateful for – but – I can't get credit for watching it on khanacademy.org. John

@Khan Academy can you make a video in how to find the degrees of a bigger hand and smaller hand in a clock and the angle of both hands cover? please 😀

I love watching these even when I'm not taking any classes.

@Khan Academy Nice video, but the second function is g(x), not f(x) as wrote at the second function's range. Putting an annotation to the video will help to not confuse the beginner's mind.

Whats the app used? @Khan Academy 😓😭

If you are a swedish read this: Läser du Matte Diskret och förstår inte Kap 3 från boken Algrebra blabla 😀 Då har du kommit rätt! Khan is awesome thanks a lot! God Bless

This is perfect! Now I totally get it! Thanks a lot~! ^_^

i have finals this week and i need serious help on domain and range of a graph and inequalities pls help!

My brain can't handle this. 🙁

very clear explaination

This helps a lot thank you your the best

Set of all positive outputs

My brain😂😩😩😩

how to do it without graph

Quick and dirty. Just my style. 😉

I'm so gonna fail tomorrow.

Isn't the last range should be g(x) not equal to 0 instead of f(x) not equal to 0?

at 6:07 is it not supposed to be g of x??

This is one of the most difficult things to Understand

I still don't understand. whyyyyy…. myghad, halp meh. (TT)

…what?

this is so confusing

Mental

Abuse

To

Humans

What if we have more variables? i.e. f = a + b + c / d * e ? how can we find the range?

Dear Mr. Khan, thank for your videos, I am attempting to learn maths by myself. The first video on Functions was funny. I read somewhere that you need funds to continue? Is this correct? p.s. Do you or one of your viewers know of an online community of people teaching themselves mathematics? G'day from Australia

Tｈis is Really Really helpful!Ｔｈａｎｋｙｏｕ

your writing is improving video by video or your software os improving

i don't understand the "thing" (x48950u43905)

I don’t even understand… lol and i have this homework r.i.p me*_* ~_~

why is the second question's range has to has a absolute real number? Isn't it all the negative real numbers are on the line too?

Somebody please answer me. Thanks

If you're seriously concerned about failing your exams, just become a police officer. Legit all you need to do is pass a medical and pass an exam that asks for you to give a logical answer. In the UK we can get paid start salaries of ~£27K and have a full scholarship in the first four years, and after seven years you can earn up to ~£45K. That's my backup 😀

Or just become a teacher, most of them don't know what they're doing either… xD

One word confusion!

2nd example's range used f(x) instead of g(x).

God,,I HATE FUNCTIONS!!!

i think my brain just imploded

What are the advantages of using that really mathy notation to answer? Is that what people use all the time in really high level maths?

Ambobo ko amp di ko pa rin gets T^T

How do you find the range of a rational function whose numerator and denominator are both in quadratic?

still confused help

in the last example you gave you said x is not =0 then drew a graph were the line passes from negative numbers to positive, why?

😘 goodbye my A+

the FIRST khan academy video that's not helpful

Why i can't handle this and i well fail tomorrow to. 1 like 1 prayer

Cool