Introduction to Functions (Precalculus - College Algebra 2)

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What Functions are and how we can determine when a relationship is a function and when it is not.

[@ronaldsanchez1341]

i turned down a scholarship to a great engineering school because i was afraid of calculus; now after watching this mans vids i am willig to go back to college although i am now 66 years old!

[@hobissprite928]

Good luck! You’re never too old to start. You will do great

[@maitumeloteemane9593]

It is never too late ....

[@cambino3696]

Hope all is well my guy!

[@christianrose2154]

I've struggled through math my whole life (21yrs) and always thought I was dealing with a literal learning disability. Turns out it was just no one cared to teach me the correct way, or cared to take time to understand that I didn't learn the same. Going to college for Electrical Engineering. I'm terrified, but so ready to work with something I findyself enjoying for the first time in my life.

[@jalinjohnson9845]

Im want to go for mechanical engineering

[@kevinguerios3001]

Never thought Clark Kent would teach me what my professor couldn’t. Thank you so much Leonard!

[@dreamcatcher9712]

😂😂 thought the same

[@felipealcarazmendez6517]

You mean Stipe Miocic

[@AnalystTosh]

@@felipealcarazmendez6517 Give it to Stipe, fire fighter, MMA warrior and fav Math teacher... what a talent!

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I graduate this spring with an engineering degree! I passed calculus thanks to you! I am forever grateful!

[@naiislam5194]

Congratulations on graduating!

[@eddylantonirina5781]

Road to graduate🔥

[@maitumeloteemane9593]

What advice can you give to a student who is about to take calculus and wishes to be an engineer like you one day....

[@TheBandFLEDGELING]

what an amazing result‼️ Major congrata...

[@charlyramirez5607]

congrats🥳 🎉

[@trademarkcr7790]

20:45 “be back in just a seconds” *ad plays with perfect timing* Me: this man is beyond reality

[@kypreadsprice]

he really is

[@Akh6one2]

I did something risky, I signed up for pre calculus for summer; completely online. Classes start on the 28th of May and I have been having anxiety, thinking that I will fail or that I am dumb. I don't know why I have these negative feelings about myself, however I just went through this entire video, bit by bit understanding this. No more funny days when I would just cheat to pass this class. My future innovations rely on calculus. I have ADHD by the way so I took several breaks while watching but I timed myself and ended up completing the entire video within an hour and 8 minutes. Including writing notes and short breaks. You explained it so well and had me engaged. Thank you once again,

[@gkwong5629]

im doing precalc in summer too ! hope everything goes well !

[@_Iam777_]

Thanks a lot, I am a 49 and i am refreshing my math to help my daughter. Leonard you are awesome.

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[@knm7218]

That's awesome ! :D

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Your daughter can learn better directly from his videos, Go live your life man.

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@@abhishekbohora9691 two people is better than one

[@beastlybombers]

Professor, I at 62 years young am returning to college to become an electrical engineer. Thank you. You are a very gifted teacher. I do support you on Patreon.

[@holybee8894]

I’m here while currently on my zoom class, but cannot understand a word from my professor. 😪 Bless my classmate who recommended this channel.

[@user-di5tv3pw1i]

You uploaded this 4 years ago, but I am learning from it right now, and you are probably the best math teacher I've ever had. Thank you so much, you are saving lives out here.

[@andrewchambers1822]

You know I always see people praise teachers (or I guess professors at this level), and they don't always deserve it. This is the best math teacher I have come across. I really think most professors, monologue from memory, whereas; this guy gets it. He really understands this stuff. From the perspective of someone who cannot learn a topic without knowing the why, this guy is gold.

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Sooo......nobody is really goin to talk about how this brilliant man is jacked And looks like a clark kent

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I think he really do his series...got it? No? Alright ...

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That’s almost the entire comment section.

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😂😂 hey let's focus

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he's hot ngl

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I was thinking the same to😂

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This is amazing, i´m currently taking precalc-alg in college online and seeing a teacher actually writing it out and explaining it in a simple way! You must be protected at all costs!

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Wonderful

[@psoren423]

I’m 43 and got my GED after the 9th grade. This prof is getting me through pre-calc.

[@poeticevolution7725]

This is going to help me ace my college courses I can tell. Already this is making more sense than when I tried to learn pre-calc in high school

[@meganrodriguez7650]

I am here getting over my fear of math. Just started your videos and I am grateful that I have a place to start with a set goal. Thanks for putting the effort to make these videos and organize them the way you did. I really appreciate it!

[@harshitkumar9672]

Best math teacher ever❤️ You made me fell in love with math. Thank you for existing ❤️

[@canman5060]

This is the best explanation of functions I have ever come across. I am a Math tutor myself.

[@aliyaaggrawal3319]

Ohh is it Actually I'm struggling too much in Maths and here is no any maths tutor where i live So can you help me plzzzzzzzz with my doubts in Maths 😥😥 Edit: plz don't mind my English 'cause it's not that good 😅

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Shut up

[@MrPowdder]

So I am going back to school for a mechanical engineer degree. Have not worked calculus in decades. Your video series is going to get me thru my Calc classes Thank you beyond measure

[@jn4198]

Im currently watching these videos a month before I start my college precalculus class.

[@Uaene]

Thank you so much for choosing to do this course, the timing couldn't have been more perfect.

[@casatkinson]

you’re an excellent professor! For the first time my mind isn’t scrambled 🥚 🍳 🥴🤗

[@queenpineasmr]

I have BAs in poli sci and economics and I bs’d my way through my algebra, stats, and econometrics. Then went on to get a masters degree in public policy (in a stem heavy CSU) and was able to bluff my way through quantitative analysis. I am terrible at math but I so desperately wish to understand it. Now I’m finally taking control and doing all I can to learn what I should’ve years ago. Your channel is a blessing from the universe! Thank you!

[@shadesofheather4616]

I'm watching this series to brush up on my college algebra that I took years ago! Thank you so much for making this understandable! :D

[@stevesmith7589]

Found this gem by mistake, at 32 going back to collage had me skeptical about calculus but i am more confident after the first lecture! Great work.

[@neelpatel355]

Thanks for the kindness and understanding that some people may not be as adapted to math than others. You’re great!

[@frens_till_the_end]

This was EXTREMELY helpful! Watching this before that start of school to review basics and be prepared for PreCal. Your repetition really allows for concepts to stick in my head so that I understand the concept and not just know the answer!

[@Yesnt38008]

These videos feel like a great example of what math teachers should do, a thorough explanation that I could easily wrap my head around

[@guilhermeolavodeoliveira5322]

Professor, you are definitely my favorite channel on KZbin. You are an excellent teacher, and there are lots of content wich I simply could not learn and did not have good teachers to help me out. Thanks a lot, you are helping lots of people around the globe. Hugs from Brazil

[@bennieholcomb4908]

I had taken Calc 1 in college but I ended up just doing programming when I should have pushed on my math and went into theoretical physics. Now I get the chance to finish my learning with the awesome Prof. Leonard. I am stoked to be able to at least understand, mathematically, the equations of the physics I so adore and wished I could have been apart of. Thanks again Prof. Leonard for your time, effort, and detication.

[@ConservativeHippie001]

Wow, this guy has such a modest way of breaking things down and explaining them. Really admirable. Great professor.

[@duyduydan]

Just started my first quarter of college since 2018 and thank you for making algebra for precalculus much less intimidating. This has easily made my outlook much warmer on my path with math in college. Thank you!

[@riffsfromtheabyss666]

Where was this professor my entire math life... Love the way you explain everything. It makes sense to me clearly! Thank you!

[@chessymajewski9669]

Hi Professor Leonard, I've always wanted to learn calculus, and now I am retired, there is more time to study than ever. It's great listening to your lectures first and then go do some examples, I feel like I'm back at school again. Thank you so much for your time and effort you put in for everybody. BTW I have Algebra and Trigonometry by Sullivan 9th edition, hope that's good enough.

[@mixtecjaguar9824]

When I have to teach this stuff, I come here to learn it.

[@xXSummonNightXx]

You're a great precalculus professor! I had to retake my precalculus mod in fall of next year, but your video helps me a lot in this subject. I really appreciate your effort in teaching us!

[@shorndrack]

Professor Leonard, I just want to say thank you for your incredible services to society. In an era where everything is fast-paced, you take your time and teach us Mathematics so we can understand what we do. Because of you, there are thousands of people doing well in their studies and making progress in their lives. Your contributions also motivate many people to do better, as we understand there are still people out there who care about our learning and knowledge. To share some of the benefits I experienced from your contributions, I did your 'Calculus I' course way back in 2017, which helped me prepare for a research project in Advanced Macroeconomics (Level IV). It was a pre-requisite for me to receive a minimum of Credit in the course to clear coursework and go to the research component of my Ph.D. program. And I was able to achieve it mainly because of your course. Now, after finishing my PhD in Economics and securing a job, I am starting over from Pre-Calculus to get a clear grasp of concepts. As a request, please start uploading videos, general, if not math. It has been seven months since you have provided us with an update. Hopefully, all is well, and your house renovation is complete.

[@DoriaRosa]

If art is human expression about the world, then teaching is human expression about the unfolding and ever-changing ways that we think and learn about the world. This art must be relational, for the ways we learn about the world cannot be pointed to and dissected, but only embodied and experienced: patience, empathy, collaboration, awe, and transformation. Professor Leonard has embraced the ability to lovingly orchestrate the subtle strings of communicating mathematics. I count myself blessed to have found him.

[@Count.Dracula46]

Politest, most passionate and most jacked professor in the history of mathematics. Love you Professor Leonard. You make Leonardo Da Vinci proud! Miss you

[@summerlane8959]

I'm taking math for the first time since 2004, and I can already tell I'm going to improve my current precalc grade! Thank you!

[@NYCGemini2000]

you make me want to hit the gym professor 💪🏽

[@MrBluemanworld]

Best channel on KZbin! What a great contribution. Wonderful knowledge, freely given - that's the way it should be.

[@georgesadler7830]

Professor Leonard, thank you for an awesome introduction to functions. This is a solid start to real and old fashion mathematics.

[@LukePaciocco]

I love what happened at 38:20 which proved precisely what you were talking about at the start of the video. thanks

[@guidofeliz8384]

Function: One input mapped to one output. This is crucial.

[@samirkumarrakshit7853]

What does that mean actually ? Can you elaborate please ? Especially mapping it mathematically.

[@noxioxz2921]

​@@samirkumarrakshit7853 f(x) = x^2 is a function cause no matter what x is it will produce one output. Therefore f(4) = 16 or f(2): 4 = or f(x+3) = x^2 +6x+ 9 etc. One input (x) will respond to one output f(x). f(2) will always be 4 but if it was say 2 or 5 then it could not be a function.

[@samirkumarrakshit7853]

@@noxioxz2921 Thanks.

[@Dazzlemebeautifully]

Ok game changer ! Thank you for these videos ! I start pre - cal Monday and this video made so much sense to me . Looking forward to watching the rest of your lessons .

[@donh7683]

This is a wonderful introduction to functions. On a recent test, I screwed up on a--what should've been an easy question--by choking on the algebra form of a function. Leonard covers the essentials in a way that is easy to understand. Thanks!

[@michaelcostello6991]

I love the amount of information you give while teaching. Thank you. You make maths interesting. I wish i had more free time to watch these lectures.

[@liamhoward2208]

The way I think about a function is that they are instructions or directions (a recipe) for or to a specific point. What value you get for Y “depends” on what X you choose. Ex: f(x)= 2x + 1 Another way to say this is: 2 times any number plus 1 will give you f(x) which is (y) Also, for beginners, functions create solid lines because X values are continuous, and/or, infinite. So the solid line actually consists of individual points that all overlap which gives the illusion of a solid line. This idea is going to be explored more when in Calculus 1 you study limits

[@pristinxx8324]

thank you! that really helps me more to understand it

[@user-om7qu6tr1h]

Thank you for the videos Professor Leonard your videos are helping so many struggling students. I really appreciate your hard work and videos.

[@mrr7870]

Need this series so bad!!! Thank you for doing this!

[@bryanmedina9808]

I’m 15 minutes in and I can’t thank you enough! This is the best teaching I’ve seen. It’s 3 AM and a part of me wants to keep learning.

[@anandx11]

cool

[@kaylenemoye]

Im so glad this popped up in my recommended! I was taking Precalc before the "lockdown" and it went downhill from there. I am hoping I will be prepared before I take it again for my fall freshman class in college. What a strange year to graduate.

[@VinnySojor]

Week 1 complete of taking Math 2 Precalc at my college. I already know these videos will help me pass. Lets Gooo!! I appreciate you Professor Leonard

[@professorfh1236]

I just turned 15 and am watching these videos to prepare for pre calc next term

[@fifivava]

I’m doing the same thing (even tho this comment was like posted like 4 months ago lol).

[@truegamer5761]

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[@CrispyApplesauce]

Good luck homie Edit: ok so I didnt look at when this was posted..

[@draperathletics8876]

Thanks for this Professor Leonard. I currently work in construction and have decided I want to re-invest in my future and the first step I'm taking is learning math and eventually trying to master Calculus. These Videos are a great helper, and I'm here with my pencil and paper!

[@studywithankitadas]

I don't have maths paper anymore still I watch your videos. You made me like maths!! Best teacher ever!!

[@akashbhullar]

The Perfect Start of Precalculus and Functions, You're the man Professor

[@haniatariq4170]

You teach in a way that calculus becomes a piece of cake and your jokes keeep me alive.

[@Align_SD]

I know this is the introduction and all, But it's been 13 years since I've sat in a math class and I left off at algebra, The way you explain things makes it so simple that even I am able to follow along. I've been scared to get back into math and after watching this I feel confident that I can make it thru it. Thank You.

[@zb5775]

You're a national treasure. Thank you for your extraordinary lectures.

[@smartiscool9748]

I’m watching these in preparation for my pre-calc class. I start in a little over a month and I was so nervous because I will be the you best in the class. I’m watching these so I feel a bit more confident. Thank you so much for these amazing videos!!!

[@bradysmith2844]

I’m 23 and am finishing my degree , and this has been super helpful I had to Take time off of college for personal reasons and this is such a refreshing step back in after some time away .

[@bri9000]

I'm in calc2 in college and find myself having to refresh on some basic stuff. I am at the point where I watch your videos for fun now.

[@ram_bam]

Professor Leonard, I love your videos. Have you considered releasing problem sets for practice?

[@talibanchristian]

I am so happy you are making this professor THANK YOU!!!!!!!!!

[@mobaidullah1544]

this was an incredible introduction to Functions. Thank you Leonard

[@harshiniharish4713]

hes so passionate lmao like hes like " I PROMISE this is how functions work" like ok sir i believe you great video tho rlly helpful a tip is to put it on a higher speed if you wanna just review. i find it helps me focos better on what hes saying cuz its so fast

[@javierv5583]

I graduated college in 2011 with a Bachelor's in Mechanical Engineering. I struggled with calculus and had to relearn trig/Algebra to survive. Years later I have forgotten it all over again and I'm reviewing just for fun. Thanks for posting these videos.

[@joviealphar3484]

My God! So brilliant way of teaching. I can't believe such people exist 😳😳😳

[@linonator]

Haven’t done structured maths in over a decade and getting a refresher. I’ve watched other videos teaching this same thing but the difference is I felt really dumb watching them. Your videos are awesome and very easy to understand for me. Thanks!

[@richardlippert8309]

I’m trying to get into school for computer science and today I found out that one of the prerequisite classes is pre-calculus. As soon as I heard that I got scared, but now that I’m watching this lecture… 25 min in and I’m sure I can figure it out. Definitely subscribing and saving this play list.

[@chriscockrell9495]

I subscribed to your channel after studying saddle-node bifurcation (which made me watch you diff eqn classes). This came up on my feed, well explained. Surprised you didn't discuss determinism. Or soda dispenser analogy in which you ask, is it "functioning." Math is simple pattern recognition. Asking the right questions goes a long way. The first time you see new material, you don't always ask the right question. You do a good job asking the right questions. The ones that show you in a way our brains can conceptualize.

[@exiawildfire7708]

oh man, so glad I found this channel, I will be starting my Bachelors degree in computer science in March after being out of school since 2006, I was nervous about going into pre calc but this overview is amazing...going to keep at this course until then to get me prepared.

[@High_priestess.]

seriously the best professor i have ever seen! I hate math but he makes it completely understandable! Thanks so much for the videos! Im watching all of them to get through pre calc then applied calc :-)

[@hassani117]

You are best math teacher ever *Professor Leonard*

[@briandang236]

I'm going back to school for mechanical engineering and you have no idea how helpful this is. Thank you Professor! ALSO, you need to do some new videos. We all miss you I am sure! Hope everything is well with your family and your life. :)

[@idreesazad]

yessss now i know what function is. Thank you professor! May you live long.

[@danielsubra4865]

....and prosper

[@momma_ravioli]

I like the way you teach, you don't make it confusing.

[@chrissya.556]

Me: Becomes a millionaire Me: Buys Prof Leonard a mansion This man is a life saver

[@thedarklord7148]

Chrissy A. Exactly my thought.

[@yash2614]

Yeah, I certainly want to meet him in person 😊😎

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You:simp

[@chrissya.556]

h j thanks... I guess 🤷🏽‍♀️

[@7moOody999XD]

@@chrissya.556 Lol ur welcome 😂

[@tahmeedmazumder6877]

You can not stop! THE internet has made you a math god. Tehcers and purveyors of knowledge roam and hunger for these things. I am glad I found you now teach me master, so that I may use my brain and help others!

[@cyancoyote7366]

Professor Leonard rocks! Thanks for your videos, love them! :D

[@rosepierce9382]

Congrats on 700K, Professor! 🎈

[@echog9110]

I can't believe it took about 15 - 20 minutes less time to understand functions with Prof. Leonard than in my classes. Thank you!

[@nick7.398]

This was a great video. I graduated from high school 2 years ago been thinking of going back to school but was scarred I wouldn’t be able to figure it out again, this calms the nerves thank you

[@fargo7018]

Bro you brought Algebra to life this is so fun.

[@ElSilvaa]

Please keep making these videos! You are really good at explaining things.

[@nicokneemepls1345]

im a college student and my pre cal class is a disaster. you saved me big time. thank you

[@DeepakPandey-qc5qu]

Amazing teaching style....Thank you Sir!

[@SuperMarvelGo]

Professor Leonard's channel is the holy grail for people who've spent countless hours studying, only to forget everything they ever learned.

[@abhishekthorat3631]

Best professor in my life

[@guitarista666]

Professor Leonard is very good at anticipating potential points of confusion or misinterpretation.

[@mariemoose8224]

Thank you for making this, I am a mom going back to college after 10+ years, this is exactly what I needed to refresh my mind for math. I can start my precalculus class with confidence for Spring semester!

[@_Anna_Nass_]

Hey same here! Good luck with your journey!

[@user-ep2lc6vw6z]

I’m literally 10 and I understand this 😂 Best professor ever🎉

[@camronel-amin3645]

Another Great Video. Very Helpful!

[@fireball0762]

the students at the HS where i sub at would listen in math if you were their teacher. Your lecture makes math seem very easy. I plan on earning enough math credits to add High School math on my teaching license. I never thought i could do math when i was younger, i think attitude is big on learning math. Thank you for making it look easy

[@philosophertiger4124]

Thank you so much teacher for your help. I am very grateful for the time and effort you put into explaining these concepts. I plan on completing the entire course and you will see a comment on each video as proof. Thankyou so much for providing this excellent education for free.

[@theoendre6484]

Spectacular lecture! Thanks professor

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You are making miracles when it comes to teaching.

[@adityadalvi9618]

And this man, just made my studies more interesting. Thank you so much!!!

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You Sir are HEAVEN SENT! GOD BLESS YOU!