Let’s do some example problems

dealing with functions and their domains and ranges. Just as a review, a function

is just an operator– let’s say this function is f; that

tends to be the most typical letter for functions– that

operates on some input, in this case, the input is x, and

it produces some output y. Or you could view it as you take

some input x, put it into your function box f, and it’s

going to operate on it and produce some y. And the set of values that x

can take on is the domain. The set of values that f can

legitimately operate on, that’s the domain. And the set of values

y that f can produce, that is the range. Now, with that in mind, let’s

figure out, one, the function definitions for each of these

problems here, these example problems, and then figure out

the domains and the ranges. So here we have Dustin charges

$10.00 per hour– so let me write that down– $10.00 per

hour for mowing lawns. So how much does he charge

as a function of hours? So let’s say Dustin charges, so

the function, I’ll call it d for Dustin. Dustin charges as a

function of hours. It depends on hours. The input into our box

is going to be hours that he worked. It’s going to be equal to the

number of hours times 10. It’s going to be 10 times– I

won’t write the times there. Maybe I can just write 10h. And, of course, he can’t work

negative hours, so we could write h is going to be greater

than or equal to 0. That’s our function

definition. If you say, hey, how much is he

going to charge for working half an hour? You put 1/2 in here, which

essentially substitutes this h with a 1/2. You do 10 times 1/2. Let me write that down

on the side here. So Dustin’s going to charge

for 1/2 hour. He’s going to charge 10–

wherever you see the h, replace it with 1/2– times 1/2,

which is equal to $5.00. That’s our function

definition. I was just showing you an

example of applying it. Now, let’s figure out the

domain and the range. So I almost explicitly say

it here, you can’t work negative hours. This number, this function,

is not defined if h is less than 0. So we could say non-negative. And we could say real numbers,

which it could be everything including pi and e

and all of that. But you can’t legitimately

charge someone 10 times pi dollars. That’s not a legitimate amount

that one can charge someone in the real world, because at the

end of the day, you have to round to the penny. You can’t actually

charge them that. So any number that he

charges is going to be a rational number. It can be expressed

as a fraction. So we can kind of take out

numbers like e and pi. So if we want to be really cute

about it, we would say non-negative. rational numbers. And these are just numbers that

can be expressed as a fraction, which are most

numbers, just not these numbers that just keep not

repeating and all of that. So non-negative rational

numbers is what he can input in here. And then the range, which is

the valid– once again, whatever number you put in for

h, remember, you can only put in non-negative values for h,

non-negative rational values for h, so whatever numbers you

put in for h, you’re going to get positive values for how much

he charges for 10h, for the value of the function. So once again it’s going to be

non-negative, and if you put a rational number here and it’s

being expressed as a fraction and you multiply it by 10,

it’s still going to be a rational number. And if you want to be really

cute, you could say any number that could be expressed

as a dollar sign. But anyway, I don’t want to get

too cute on this problem. I think you get the idea. Let’s do more problems. All right. Here we have Maria doing

some tutoring. Maria charges $25.00 per hour

for tutoring math with a minimum charge of $15.00. So Maria charges as a

function of hours. So she’s going to charge

$15.00 if you don’t get enough hours in. So at $25.00 per hour, in order

to make $15.00, you’ve got to work 15/25, so

it’s 3/5 of an hour. So if her hours are less than

3/5 and greater than or equal to 0, she’s going to

charge $15.00. Because if she only worked 1/5

of an hour, the bill would have been $5.00, but

she says she has a minimum charge of $15.00. So she’s going to charge $15.00

up until 3/5 of an hour, or 36 minutes. And then after that, for h

greater than 3/5, she’s going to charge $25.00 an hour. She’s going to charge

25 times h. So that’s her function

definition right there. Now, what is the domain,

the domain for Maria’s billing function? Well, once again, this

is only defined for non-negative numbers. And by the same logic we used

here, we got a little cute. We said, oh, she really can’t

charge e hours or pi hours. That’s not realistic

for her to measure. Everything’s going to be

expressible as a fraction. For example, even 5,

if you say 5 hours, that’s still a fraction. That’s 5 over 1. So let’s say non-negative

rational numbers. And what’s the range of her

charging function or her billing function? Well, it has to be

at least $15.00. No matter what, she’s going to

charge $15.00, so it’s going to be $15.00 or more. So it’s going to be rational

numbers greater than or equal to $15.00. There’s no situation in which

she will charge $14.00. There’s no situation

in which she would charge negative $1.00. Everything’s going to be greater

than or equal to 15. So now we have these more

abstract function definitions, so now can stick really

any number in it. Well, we’re not going

to deal with complexes, any real number. But let’s see if they limit

it a little bit. So we have f of x. Let me rewrite it. f of x is

equal to 15x minus 12. So what’s the domain here? What’s our domain? Well, I can stick any real

number x in here and I’m just going to multiply it by 15

and then subtract 12. I could put pi there. I could put the square

of 2 there. I could put e there. I could do all sorts

of crazy things. So I’m going to say

all real numbers. Any real number I put in for

x, I’m going to get a legitimate output. And then what is my range? Well, once again, this can take

on any value out there. I can get to any negative

value if I make x negative enough. I’m going to subtract 12 from

it, but I could get to any negative value. I could get to zero. I could get to any

positive value. So this is also going to

be all real numbers. Now we have this function. It says f of x is equal to–

they forgot the equal sign– 2x squared plus 5. So what is the domain? What are all of the valid values

for x that I could stick here? Well, I could stick

anything here. I could put e there. 2 times e squared plus 5. I could put a square

root of 2 there. I could put a negative

number there. So I could really take

on any real number. All real numbers, positive or

negative or otherwise, or even non-rational. And then what is

the range here? And this is interesting. Because no matter what number

I put here, this value right here is going to be

greater than zero. This is going to be, right

here, non-negative. Even if you put a really

negative number here, you’re going to square it, so it’s

going to become a really positive number when

you square it. So this expression right here

is going to be non-negative. So if you take a non-negative

number and you add it to 5, you’re always going to get

something greater than or equal to 5. Worst case, this is a zero,

and then you get a 5. But if this is either slightly

less than zero or slightly more than zero, you’re going to

get values greater than 5. So the range is all real

numbers greater than or equal to 5. Your f of x could never take

on the value zero. There’s no way to get zero. Problem 5. They have f of x is

equal to 1 over x. So, once again, our domain. Well, we could put any

value in for x. Any real number will

work here. We’re just going to take

the inverse of it. So this is going to be all

reals, all real numbers. And then what is the range? This is interesting as well. Because no matter what I put

here, I can put anything here. Oh, actually, let me be

very, very careful. This is not defined for

x is equal to 0. All real that are not

equal to zero. If I put zero here, I’m

going to get 1/0. It will be undefined. Let me write that down. I almost made a mistake.

f of 0 is undefined. And that literally means

we don’t know what to do with the zero. We have not defined a way

to operate on a zero. So f of 0 is undefined. So the domain is all reals

except for zero. And now what’s the range? Well, once again, we can take on

any value here except zero. We could try to approach zero. If x got really, really, really,

really, really large, then this will approach zero. If x got really, really

negative, it’ll approach zero. But in no circumstance will we

actually ever get to zero. So once again, all reals that

are not equal to zero. All real numbers except

for zero. All right. Problem 6. What is the range of the

function y is equal to x squared minus 5 when the

domain is– so they’re defining the domain. They’re restricting it. They’re saying the domain is

just the numbers minus 2, minus 1, 0, 1, and 2. So this function definition

really should say f of x is equal to x squared minus 5 for x

is equal to– we could write any member for x is a member

of the set minus 2, minus 1, 0, 1, 2. It’s not defined for x being

anything else to that. You don’t know what

to do with it. You can only apply it if

x is one of these. So what’s the range? Well, the range is all of the

functions, all the values that f of x can take on. So the range is going to be the

set of values that we get when we put in all of

these different x’s. So let’s try this first

one: negative 2. Negative 2 squared minus 5. That’s 4 minus 5. That’s negative 1. Let’s do negative 1. Negative 1 squared minus 5. That is 1 minus 5. That is negative 4. You have 0. 0 squared. So f of 0, you have 0 squared

minus 5, that’s negative 5. And you have 1. 1 squared minus 5, that’s

negative 4, so it’s already in our range. Then you have 2. 2 squared minus 5, that’s

negative 1, already in our range. So that’s all of the values that

this function can take on, given that we’ve

restricted it to only these inputs.

why 27 dislikes, so many haters out there!

This guy is so helpful, helped me through my A-Levels and I can actually learn maths. Thank you, and keep up the good work

Sal just got me through my algebra 2 test. Best math videos ever

Thanks a lot – really helped me out.

"A really positive number"… hahaha. =)

omg

I am so glad that Jason and Khan Academy has put this all online since the college I go to sold me the wrong book twice, Thank you

at first, I would have thought that he had made a mistake when he said that the domain of f(x)=1/x is all real numbers.. cuz I was sure that zero cannot be put because it would be undefined, but then he found out and im like SNAP……… :)))

At number 5 the domain should be all real numbers unequal to 0.

Edit: nvm i was panicking early and typed before he saw his mistake

41 dislikes? haters gonna hate!!

i thought a function can not have 2 answers? as in one domain can not have two ranges?

for question 2:

shouldnt Domain= h =or> 3/5

is answer 2 correct? if somebody stays for 4/5's of an hour, do they get charged $25 (ie a minimum charge of 25 dollars for less than an hour?) they should get a base charge of $15 and a proportional amount for the amount of time 3/5 > 1 for the first hour. as it stands the minimum charge is $15 for less than 3/5 or $25 dollars for anything more; up to an hour. Thanks for the video btw, they're awesome 🙂

Question 5: would the range be equal to or less than 1 seems x is the denominator?

lol how much he uses cute

For question 2, I'm trying understand the logic behind this. If Maria charges 15 dollars minimum that would mean that M(h) = 15 0<= h <1 since she charges 15 dollars until an hour has been reached, right?

What is y=x squared minus 5?? Please answer!!

I was sick yesterday and now my teacher refuses to help me review or show me the notes so yeah…

Thanks Khan!

Namaste very good excellent job

i still dont get it

sal u neeed to explain it to me like im a five year old or something

anyone out there

. . . I still don't get it. Everything went in one ear and out the other. Can someone else explain it to me? Like, try explaining it to me like your explaining it to a 10 yr old, please.

3:07 what's the symbol for rational numbers?

You're a terrible teacher you confused me even more!

I can't see the problems sighs and I literally hunt your videos…

Delete your video please

I can't still get this right. Oh gosh

Still dont get diffrence of range and domain

I have been through the entire of Calculus 1 and 2. Still manage to forget this 😅

This might as well be a foreign language. This makes no sense😩

I don't get how people do not get this math. Its basically learning something like chemistry, physics…

TTTTTTTHHHHHHHIIIIIIIISSSSSSSS. iIIIIIISSSSSSS. SSSSSSSSPPPPAAAAARRRRTTTTTTAAAAA. ,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!L!!!!!!!

Idk why but I seem to be able to not focus in class since my teacher is so monotone.. I just need something like this to help me comprehend so thanks

ima cry

nkich

Thank you

Now how would you work with infinity when it comes to domain and range?

That's what I'm confused about.

Thaaaank you

Sorry I don't speak math

Isn't -2^2 =4? Then y does my calculator read -4

Wow…You are much better than my tuition teacher…

I'm studying for an entrance exam into grad school and I can't even do this shit…

Is domain and range simply input and output, respectively? Why are they called domain and range?

Thanks really appreciate it

ima be tellin the truth. this is way better than goin to tutering after school!!!

this is hard

8:07 That's what she said

(I'm so sorry)

What! Maria is so greedy. My maths tutor charged $15.5 a month, averaging 15-17 hours a month

I can't see it ,_,

This is a terrible video, all the examples have domains of pretty much anything. What use is that? Need examples of other types! Disliked

The problem with Khan Academy is the pacing it also goes at a too slow pace.

It's so hard

you teach this so easy and understandable, meanwhile it sounds like my professors speak another language when they teach this

i'm not sure but the domain in no. 3 is (-12,+ infinity)

So many mixed comments saying it's confusing or absolutely amazing. WHICH ONE IS IT AAHHH

your way of teaching makes me sick

But both the domain and the range need to be inequalities…? Not just cop out and say "all real numbers"… Right…?

8:27

This was the best explanation I ever searched. Thanks very much

i love khan academy they teach me lots of maths THANKS!

For question 4, why can't the domain be lower than 5?

why am i so confused

I don't understand

Who needs a math tutor when you have Khan?

Couldn't have explained that better

Can someone explain How Domain & range are Written in sets Like form

Like domain: R-[-2,2)

I would appreciate it if someone would help me out

KFUPM

Aswm sir

Stop saying cute

Good video, but why do you say cute so much?

Marvelous explanation. Example number one gave me a practical view, just what I needed.

I HATE FUNCTIONS! They make it complicated for no fkn reason!!!

How cute

why cant we put complex nors in functions?

When he says as a function of ….. it confuses me

bruh

Sir how to find the range of x^2-9/x-3

9:22 "There's no way to get zero"

Audrey Hepburn calls in the background, "nothing is impossible.."

How do you get the range of a rational function whose numerator and denominator are both in quadratic form? Without using a calculator how do you get the range?

What does 0 have to do with the first one?? I'm just dumb 😭😭😭🙁🙁. Of course he doesn't work 0 hours but what does it have to do with Duntin charging $10 per hour.

I hate 8th grade

Yaaaaaaaaaawwwwwwwwwnnnnnnnnnnnnnnnnnnnnnn😞😞😞😞😞😞😑😑😑😑😑😑😴😴😴😴😴😴😴

Good job 👍 tysm

What if you had y=x^2 ad you had to find the domain and range of just that equation.

Lol I'm so dumb I can't understand this.

I’ve come to the conclusion that I am stupid. Math does not make one ounce of sense to me

The 226 dislikes are those who don't understand this, one of those may be me.

This was really useful, thank you!

Question no. 5: can I say that the range is smaller than or equal 1 because it will never be bigger than 1 actually cuz this is a fraction?.

I'm confused help

Thank you so much!! This is so helpful for me.

This was amazing! Was able to understand each and every bit of domain and range now. Thanks so much Khan Academy!

And for those peeps who don't understand this, you need to go and learn what functions are first. These are basics you need to master

You explained 90% of the concept in the first few seconds of the video

Well i think you teach really very well, i wish you were my teacher cuz my teacher said that it's too complicated for you kids so just memorize it and bcz of him now i hate maths but if you would've been my teacher i would be an expert by now😄😄👌👌

Sir you are really confusing us with so many operations

you explained it better than my teacher

I feel so bad because I still don't understand huhuhuhuhu.

Really ,this is so helpful

Just because a guy helped me for my exam without any fees, I'm gonna teach kids for free.

In number 2, should the condition for $15 be 0<h<=3/5?? Since you will not get paid if you work 0 hour regardless of the given situation. Please let me know if I'm correct or not , I would appreciate the answer😊