Back to Algebra: What are Functions?

[@terukun5645]

I'm on my third day reviewing math from the basics until this episode and everything has been a smooth sailing journey thanks to Professor Dave. He explains everything crystal clear and I swear to god, his organized playlist of math video lessons is the best thing that you could find out there for learning mathematics and it will definitely guide you into fully knowing the basics and other info you missed out on or have already forgotten during your high school years (or lower grade level years) of studying mathematics. Dave, I wanna give you a big big hug and a big big thank youuuuuuu for creating these contents! They were very helpful to me, especially during this time when I will soon be taking college entrance exams. Once again, thank you soooo much! And I wish that more people would come to this wonderful and helpful channel!

[@ahnafalnafis]

I was pretty dumb at Mathematics. My friends sometime laughed for it. In high school, I was getting worst marks in Math. I found out where was my problem and it was in foundation. Then I looked up into KZbin so that I can find a tutorial or course where from I can learn the basics of Math and then head to advanced concepts. Then I found this genius' channel. First, I thought that it'd be a simple tutorial like we always find in KZbin. But I started to follow this series and I'm glad for that. I found an elegant beauty in Mathematics and started to love it. Math seemed interesting than ever!!! I also started to get better marks in Mathematics! I don't know how I can thank you. Thank you soooooo much for making such contents for us!!

[@tazepatates4805]

Math is like meth, once you do it properly you can't get off it. (I never did meth tho)

[@ahnafalnafis]

@@tazepatates4805, Exactly, bro! Now I love Math this much that I don't need breaks for 4/5 hours straight when I'm making love with Math. Also, apologies, my English was bad at that time. Now it's kinda good and I edited the comment to make understandable 😅

[@TheKids422]

Thanks, you deserve the nobel prize for the work of humanity education.

[@DoubleQCubed-d7o]

the *upgraded* version 0:00 It’s Professor Dave; let’s learn about functions. 0:09 The standard practice of teaching math in school is to teach a year of algebra first, 0:15 then a year of geometry, and then do another year of algebra, which is often called Algebra 2. 0:22 Since we’ve just done a lot of work with geometric figures, we are ready to come back 0:27 to algebra and expand our understanding of this subject. What are Functions 0:32 The first thing we want to do is comprehend the idea of a function. 0:37 A function is something that relates two quantities, an input value, and an output value. 0:44 So we can think of it as a little box, and when we insert input values, it spits out 0:50 the output values, in a way that depends on what the function is. 0:56 Let’s say at a particular store, everything is thirty percent off today. 1:02 That means to find the new price of an item we would multiply the original cost by 0.7, 1:08 because that will give us seventy percent of the original price, which is the same thing 1:13 as thirty percent off, since one hundred minus thirty is seventy. 1:19 The function that we would therefore use to compute the sale price of any item would be 1:25 F of X equals 0.7 X. F is the function, and, the X in parentheses, which is read “of 1:34 X”, means that the function will operate on any X value that you plug in. 1:40 If we plug in one dollar for X, the function spits out seventy cents, so the F of one equals 0.7. 1:50 F of two equals 1.4, F of three equals 2.1, and so forth. 1:58 We could make a table like this, and we could even graph the resulting relationship. 2:03 So everything we used to do with equations where Y depends on X, we can do the same thing 2:09 with functions. 2:10 The only difference is that with functions, for any X value we plug in, there must only 2:16 be one Y value. 2:19 For example, if a function tells us the sale price of an item if we plug in the regular 2:24 price, we should get only one sale price. 2:29 For that reason, for a graph to represent a function, it must pass the vertical line test. 2:37 Since a function can only have one output, or Y value, for each unique input, or X value, 2:43 then if we move a vertical line across a function, it will only intersect that function at one 2:49 point, wherever the vertical line may be. 2:53 If the vertical line intersects at two or more spots, it is not a function. 3:00 That would be as though we plugged in the regular price of an item and got back two 3:05 different numbers for the sale price. 3:08 We can, however, have the same output for multiple inputs, which is why different X 3:14 values are permitted to produce the same value for the function, so there is no horizontal 3:20 line test necessary. 3:23 This line is a function. 3:25 This circle is not. 3:28 This curve is a function. 3:30 This curve is not. Domain Range 3:33 We can also describe the domain and range of a function. 3:37 The domain is essentially all of the values that can be plugged into the function. 3:43 So if we have something simple like F of X equals two X plus one, the domain would be 3:49 all real numbers, or negative infinity to positive infinity. 3:55 All the function does to the input is double it and add one, so any number will work. 4:03 But there are examples where there are limitations to the domain. 4:06 Let’s say F of X equals one over X minus two. 4:12 We can’t divide something by zero, so X can’t be two. 4:16 If X was two, we would get zero on the bottom, and the function would be undefined, so two 4:22 is not part of the domain. 4:25 The range, on the other hand, is all the potential output values, or essentially the values that 4:31 the function can possess. 4:33 Again, something simple like F of X equals two X plus one will yield a range that includes 4:40 all real numbers, extending to infinity in each direction. 4:45 But if we have something like F of X equals the absolute value of X, then the range is 4:51 now limited to values greater than or equal to zero. 4:55 So while the domain is very frequently all real numbers, the range has a tendency to 5:00 vary quite a bit depending on the type of function we are looking at. 5:08 We will also sometimes want to identify the zeroes of a function. 5:13 These are the input values that make the function equal to zero, which we can identify graphically 5:19 as the X-intercepts. 5:22 Wherever the function crosses the X-axis, that value of X is producing a Y value of 5:28 zero, so that is a zero of the function. 5:31 Again, different types of functions will have different numbers of zeroes. Evaluation 5:37 Before we look at different types of functions, we need to make sure we know how to evaluate 5:43 a function. 5:44 For the function three X squared minus five X plus two, what is F of two? 5:51 Well, let’s rewrite this but with two in the parentheses instead of X. 5:56 That means we are evaluating F of two. 5:59 We then also put a two instead of an X wherever there is an X. 6:04 Two squared is four, times three is twelve, five times two is ten, so twelve minus ten 6:11 plus two equals four. 6:14 F of two equals four. 6:18 Now that we know what functions are, let’s check comprehension. 💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶💴💯💴𐂀𐂀💶💶

[@nalaroe3646]

I love that function machine! It's so satisfying watching everything roll into it and come out converted on the other side! 😃

[@basketguitar9768]

Just like eating food

[@GuidelinesViolator]

@@basketguitar9768 U bugging

[@zarachauvin4231]

Oh my gosh I'd looked at so many other videos and none helped but this made me get it right away! Thank you so much - so well explained

[@raheennock9502]

Glad I found your channel. I'm taking college Algebra/Trigonometry class. You break things down very easy💪🏾💯. Thank you

[@V1CT0Rc001]

I'm sure you've heard this quite a few times already, but this video just cleared up almost ALL of my confusion about functions. For some reason, I just could never understand the explanation provided by Time4Learning. THANK YOU PROFESSOR DAVE!

[@1Dama1-]

Hello,Can u write a summary about the main ideas of this video.

[@chiklachikla7641]

@@1Dama1- he explained the idea of function which is a relation beetween two variable (let say x and y) for every value there is a unique (only one) corresponding value of y a function of x is represented as f(x) all the possible that x can take is called the domain all the possible that y(or f(x)) is called range in order to know whether à graph is function you do the vertical line test (you draw a vertical line) if there is more than one corresponding of y it is not a function while the opposite can happen and he explained thé concept of zéros of à function which are the values of x when f(x) or y equal zero; and that pretty all there is to it

[@Ranbo5]

Love this guy. Unlike some "teachers" he doesn't assume you already know what he's talking about.

[@shubhamsingh3635]

very good explanation.

[@farhanahmed2508]

First like! I wonder which heathen left a dislike as soon as the video was uploaded. Anyways, make sure that you do Pharmacology soon, Professor! Cheers!

[@ryanjamesconiendo4429]

Thanks professor Dave as always. . I've watched explanations about functions, yours is the most comprehensible 💪 with a great animation and examples as always

[@Electodiminuendonulecresedo]

Dave,not hyfuluten(spell wrong),becouse i find out that chinese cousese are not as clear as you do,i learnt English!you are a good teacher😇

[@zilog1]

This made my computer programming class so much easier to understand :p thanks

[@utpaljha4357]

Whoa!!! Lovely professor lovely....

[@simprod9066]

thank you very very much teacher you are great person

[@rnbhari5426]

Nice video, thanks

[@annieli3258]

Thank you!

[@atashidey9414]

but the square root function yields negative and positive values, is f(x) = sqrt(x) not a function then ?

[@Wutheheooooo]

he has something wrong, the sqrt(x) is not the same as √x, √x prefer the positive value and when we want both, we may write sqrt(x) or ±√x

[@jose_w3128]

Hey, Dave, would you consider f(x) = √x a function? I've been watching some of the videos on your Math playlist, which have been coming in handy for my studies, and it caught my eye that f(x) = √x appears as a function when I google it, even though you've shown us in the video about square roots that √9 = ±3. What's up with that?

[@justmehi8700]

Hey, Functions take the non negative values only, so only 3 would be valid for the square root of 9 as a function

[@pramitkumardatta2771]

I think he skipped that here (which almost never happens without a clarification comment later on, so am I wrong Dave ?). When he says Functions cannot have multiple outputs for same x, Roots CAN be a function, and there is no standard where a negative cannot be a function value, but I could be wrong. I'm just hoping Dave is baited into replying here and I can thank him WHOLEHEARTEDLY for the playlist (and other playlists).

[@Marcus-et2if]

Typically there is an agreement in maths that the root of a number is always positive. The negative solution will be disregarded, even if it is technically also a solution. Thats why functions with roots can exist. Another reason is to harmonize roots of even and uneven order.​

[@Wutheheooooo]

@@pramitkumardatta2771 √x refer the positive value i believe, when you want both, you will write it as sqrt(x) or ±√x

[@DoubleQCubed-d7o]

Gran video La profesora Dave explica

[@DoubleQCubed-d7o]

I I'm in UK by the way

[@satyamc]

Sir make a video on function for class 12 based on indian school and iit jee

[@LoveIssa-mj9em]

Hey in comprehension we have first question is 60m^3

[@muhammadaftabalam7412]

Sir make some video on calculus too

[@tGoldenPhoenix]

Done this lesson.

[@1Dama1-]

Hi,Can u write a summary for this video

[@hasibulalamprionto2010]

we've come so far but still didn't get any lecture about our most renowned formula of algebra: (a+b)^2=a^2+b^2+2ab ..........??

[@mustafasaber5076]

Domain Range 1

[@PasionDeArgentina]

At 6:15, shouldn't it be zero, following Bodmas , we will have addition first (10+2), and then Subtraction , 12-12, yielding zero ?

[@k1tooy_m]

It would be -10+2 which will equal -8 then you do 12-8 which is equal to 4. The order of operations won't matter of you're only solving using addition and subtraction. They have the same value or importance in the PEMDAS

[@Wutheheooooo]

@@k1tooy_m but 12-10+2 is 0. and when they have the same importance, you do them left to right. Let's forget pedmas and use parentenses for every operation 12 - 10 + 2 is equal to ((-10 + 12) + 2) now you can do what ever order you want. It's how math work

[@lucasthompson1650]

Chicago (original Macintosh menu font) lives!

[@thejackal5099]

So this is like a function computer programming. Interesting.

[@pinklady7184]

I have learned "what is not a function."

[@SankalpaSatyal]

Dope😊

[@SankalpaSatyal]

Dope😊