How to Evaluate Functions (Precalculus - College Algebra 3)

[@horizonbrave1533]

"Can you take zero slices?" "Yea it's called a diet, you don't eat the pizza".... My god Leonard you have so many classic lines..

[@hiroshi3273]

I did not anticipate that line. 😂

[@qariel8103]

this was such a great analogy haha, had me chuckling

[@aya5468]

I laughed because it made sense. 😂

[@RStell-wt5qr]

I've never contributed to anyone's patreon, I'm very poor. But I'm willing to sacrifice a meal or two to support this kind of content.

[@TheEndernal]

How am I the only person who liked this comment so far should get more likes

[@LunarKR]

kind of you to do that, but thats a double-edged sword. i agree with donating to professor leonard's patreon, but it's not worth missing a meal over, i mean, food is more of a need than this.

[@arpanchakraborty908]

True​@@LunarKR

[@Positron_6.022]

@@LunarKR is right, don't skip meal buddy

[@elitevortex6098]

Maybe you guys are too late

[@jacobrzeszewski6527]

KZbin is such a valuable tool for my learning. I have ADHD and being able to rewind back to go over a section I blanked on or missed is so useful.

[@AmmarKhan-zo3fy]

Attention deficit hyperactivity disorder right?

[@thatomofolo452]

​@AmmarKhan-zo3fy it's not like a learning disability right 😮

[@ROTHSTEIN01]

​@@thatomofolo452 It is a learning disability

[@arsdwalos6840]

You quickly became my teacher that I never had. Thank you from my bottom of my heart

[@MathswithMuneer]

Yes he is doing a great job. He inspired millions. That is why young passionate like me are in this profession as well. :)

[@me-tubemath7927]

Today is teacher's day in india. Mr. Leonard your teaching style is great. Prealgebra Intermediate algebra Cal: 1,2,3 TTP VIDEO Differential eq. ...all are very helpful my life. HAPPY TEACHER'S DAY mr. leonard.

[@EKDupre]

A very kind sentiment, multiplied by 60

[@christinadsouza5661]

Is this for 11th grade or for college students the video

[@me-tubemath7927]

@@christinadsouza5661 yes

[@Nishant33p]

@@christinadsouza5661 pre calculus and calculus 1 is a college course in US but is taught in india in 11th and 12th. In 11th algebra 1, algebra 2 , trigonometry is taught which is equivalent to pre calculus and in 12th grade calculus 1 is taught. So if you are in 11th grade in india you could watch this series.

[@laundrysauce1649]

@@Nishant33p depends on the student. All highschools teach pre cal and cal 1. Generally you take pre cal senior year, but you can be a year ahead and take cal 1 senior year aswell.

[@PaquiCamus]

I have just subscribed to your channel. I am a retired scientist(hydrogeochemical), a chemistry degree of about 50 years ago. I did calculus and algebra but have used just a little. Now, I have to help my grandkids to get into University and they want to be engineers and scientists. No problem to help them in chemistry and physics but Advanced Maths to prepare them for Calculus is a great pain. I have been avoiding it until now. Universities in Australia have Advance Maths plus Extension as a pre-requisite. Consequently, I have no choice but to review pre-calculus and calculus. So far I have really enjoyed your presentations and advice: "Do not do in your head but worked out in the paper" Great advice! I like the way you teach and present the topics. I am catching up and teaching my grandson for his preparation. Thank you very much, Professor Leonard. Still have 3 granddaughters and 2 grandsons waiting their turn.

[@allankafue455]

The taylor swift blank space just gets me all the time

[@zachallen6256]

You have had impacted my life in the best way. I thought I disliked math, and avoided degrees that required it. Now I am getting the chance to give it another shot, and I have never felt more confident. I have also never had such a great teacher.

[@eddieolsen100]

did you get your degree

[@Azud8]

I am an Arab, but I follow you because of your excellent explanation and lesson. I am so happy that I finally understood the lesson.Thank you very much ❤

[@Rainsong777]

May the Lord bless and keep you for helping so many students across the globe to ENJOY Mathematics again !!! :D The world needs more teachers like you, who actually teach in a way that doesn't BORE students, and also... you don't speak in a way that "looks down" on the students ! All of us here on KZbin appreciate your effort and good work !

[@Rainsong777]

09:01 ... what I tell my students from Grade 7 and up... "chuck it in a bracket" !!! :D It just WORKS !!!!!!!!! No mess, no fuss... keeping it simple and VISUAL... even without a calculator !!! which too many students have grown to be literally "attached" to ! :(

[@Rainsong777]

11:33 ... writing the "input-output" as an "ordered pair" of coordinate points is also something that we can easily forget... and that is actually what the whole exercise is about... finding the input/output "connection" ... the (x;y) of the function at that specific point :D

[@daviandrade012]

When you realize you're watching professor Leonard's videos like you would watch friends, that is, just for fun. Best professor ever!!

[@jackie1710]

At 31:15- Best explanation of dividing by zero I’ve ever heard. I needed this in grade school. Lol .

[@douglasgorney]

I never ever ever got this stuff in junior high and high school, and when it came to college, I didn't even try. Wish I'd had you as a teacher then; it might have gotten through my thick skull. Have always felt incomplete regarding math, so I'm finally going back with the intention of getting to and through calculus. Thanks so much for the help!

[@megugu2155]

same bruh. my teachers back then never was interested on math nor do they bother to tell you what it is for. you just end up going "if i dont know whats this for then why should i learn it?" your entire school life. thank the heavens i learned the purpose of math just before college. am learning just like you amen.

[@Peter_1986]

I got extremely stuck on Trigonometry during an optional math course in high school about 20 years ago, and dropped out of that course. But I did find math really fun, so I actually started reteaching myself as much as possible from all the previous math courses from junior high school and onward, and made absolutely sure that I understood exactly what I was doing, and this then allowed me to finish Trigonometry, and at this point I have covered all of Professor Leonard's playlists, as well as studied some partial differential equations and fluid dynamics at university. It's really a lot about regularly reviewing stuff, and keeping an eye on both old and new material. It's also a good habit to try to derive laws and formulas on your own, since this shows that you know where they come from and how they work, and also guarantees that you always have access to them.

[@DirewolfMemento]

For those confused as to why (x+1)^2 = x^2 + 2x + 1 Here is why it is since he brushed past it a bit. the expression (x+1)^2 = (x+1)(x+1) meaning (x + 1) times itself. Which means you would take (x)(x) and get x^2 and (x)(1) = x and (1)(x) = x then (1)(1) = 1 If you simplify you would combine x + x which equals 2x Then your answer would become x^2 + 2x + 1

[@StarPlaysRoblox13]

Tysmm I needed this

[@ernieparchment2195]

Thank you professor. I am taking pre calculus after being out of school for over 20 years. You make learning math easy and enjoyable and I can see your love of teaching. Best professor ever😊

[@nihilnothing]

I'm taking Pre-calc in 10 days. I'm pretty nervous to dive into math again, so these videos are *much* appreciated.

[@supersonik76]

How did your pre-calc class go? I'm doing the same in 2weeks.

[@nihilnothing]

@@supersonik76 Hey, it went great. I got an A. It was a lot of math, but nothing impossible. I had to remember how to divide fractions and stuff. My class covered domain and range of a function, x and y-intercepts, slopes, y-mx+b, point-slope, asymptotes, that kind of thing at first. Then we got into trigonometric formulas, cos() and sin() stuff, finding the sides and angles of a triangle. It's pretty much all graphing functions and defining their characteristics with a bit of geometry thrown in. If you're watching videos like this to prepare I'm sure you'll do great.

[@supersonik76]

@@nihilnothing thanks for the info, it sounds like a lot of work but im sure ill be find.

[@airmanfloyd719]

@@nihilnothing So if I learn his pre cal Playlist. I should do really good if I take pre cal?

[@nihilnothing]

@@airmanfloyd719 It'll definitely give you a better chance. If you have the time I'd recommend it.

[@GammaFields]

Thank you, Professor! This is roughly where I left off in HS almost 8 years ago. Watching these has been the perfect refresher to prepare me for my transfer degree at my C.C in April. I've been bouncing around between different lectures, and yours is the one that's really made it click/set in. You've given me many "ah-ha!" moments, and Im looking forward to working through the rest of your video series to further prepare myself.

[@krystinkaminski9014]

I would have never have made it through any of my math classes without you! Its because of you I was able to teach myself all the way through statistics and skip 4 semesters of math. Your the best and I recommend your videos to anyone learning math. Everyone who took my advice all agreed that you are amazing. I wish you taught financial accounting and investment analysis too!

[@dafquix8977]

my pre calc professor right now is so fast and I feel so dumb when I cant understand her, but this guy explains everything so well, thank you so much, you're a lifesaver professor

[@funlearning5506]

Same here my teacher is also teaching in a very fast way

[@erijaqrab5480]

I am returning to Algebra and functions which I did way back in 1962 and 1986, I wish my teachers could have seen these videos, probably then Math would not have given us chills. I am sure I will watch all the presentations because I must take calculus for my research.

[@ehab6605]

I recently transferred universities and lost my math credit and as a junior this is the hardest class that I'm currently taking but these videos have such clear cut examples and explanations that it honestly dissolved all my worry, thank you for making these videos and they will definitely assist me this semester

[@dmorgan0628]

Thank you for all your hard work Mr. Leonard, sadly I've regressed and haven't taken calculus' courses in several years and would like to get back into it. I landed a federal job and worked hard in that field rather than completing a degree but would like to bust rust and at least try to finish what I started and pass the difficult math courses I failed at. Here I am and will probably donate to your patreon once I figure some budget stuff out. Thank you again Professor Leonard for all the hard work you continue to do (at the gym of course ;)) Keep up the gainz! Dan

[@trenckens]

I got really excited and said Voila- when you explained this output 2-output 1/ change- made the abstract stuff a lot more real

[@hjkjy4602]

This made me think that i'm probably not hopeless in math, thank you so much Prof Leonard!

[@dayanaracastaneda9731]

Professor Leonard you are truly a life saver, and one of the best professors I’ve ever encountered. My college professor currently always manages to make simple concepts complex and you make everything easy. Thank you very much.

[@MariniaGroup]

This is probably the best math instruction I have seen on this topic. Excellent teaching.

[@adityadalvi9618]

"This is what we mean by evaluating functions"... "You can put anything you want".... "You can put a smiley face if you want"... Just amazing teaching :) Thank you so much Professor Leonard for making Mathematics exciting to learn.. Really grateful

[@MathswithMuneer]

You are doing a wonderful job and you inspired millions, that is why young teachers like men are also in this profession. Together we will make world without fear of math.

[@daonguyen6320]

That explanation on the usage of the difference quotient was mind-blowing!

[@TheDhdk]

That bit about the difference quotient at the end was absolutely incredible!

[@cristhiangarcia5991]

Dear Professor, I wanted to express my heartfelt gratitude for all of your dedicated efforts. Your hard work has not gone unnoticed, and I truly appreciate it. I am particularly fond of your teaching style, and it has greatly contributed to my love for learning. Thank you once again.

[@Caribbean_King]

I never knew or understood the why behind the math or what it was saying to us through numbers, you explained that perfectly! Thank you! I've always hated math teachers telling you to do these problems but never breaking down the concept's logic and reasoning. Math is a language and there's more to it than just numbers on a page/screen.

[@surojpaul14]

Nobody teach me such kindness and gentle way,, you're a hero 🦸‍♂️Leonard

[@sabrinanaser9327]

This was so helpful omg. I took precalc & calc in high school and I did well, but when I got to college I realized I needed to retake precalc if I wanted to do decently in calc (my required class). thanks for helping me build a better foundation for my calc class next semester! Very grateful I retook this class--- i know a lot less than i thought i did.

[@mikewalker6770]

I made this comment before but because you have sooo many awesome videos, maybe, you didn’t see: I would love to hear your thoughts on simply teaching -not just math- but your theory of learning and teaching. I know a lot of people who work in numbers but MOST cannot teach it. Not only can you do but you can also teach. So, what’s your ideas around learning, memory, and teaching. Like a masters video in teaching abutting. You’re awesome !

[@liljay3019]

I come here just to like your vid! Thanks for the all the help in my Calc 1-3 courses prof. Leonard!

[@MathswithMuneer]

He is amazing. no doubt :)

[@abiygirma3354]

Everything you explain seems simple and easy yet, it is a very basic and firm foundation to Math. Thank you, Professor.

[@Knawlege]

This man makes math actually enjoyable

[@CoolMachination]

Wow all these free, Thank you Prof Leonard!!!

[@maris6843]

Thank you for this series. I love that you're not just telling us what to do, but why we're doing it. I love the pizza and diet analogy. Relating math to food always makes it more understandable haha

[@heydyvex]

YOU are literally SAVING my life Professor, Please keep this going

[@OkaniRikhotso]

Thank you so much proffessor Leanord! You have made my life easier! I recently wrote a math test and I do not feel confident about it but now that I have discovered your channel I have no doubt I am gonna ace the next test! THANK YOU SO MUCH!!!!!!

[@robpatterson2861]

You are too cool dude! I'm 56 and loving your videos! Your Taylor swift joke was hilarious!

[@radhikamadan376]

HAPPY TEACHER'S DAY TO YOU !!! Sir , Thank you for helping me to break free from my fear of Mathematics . You have not only made it possible for me to be able to understand the concepts in a really better way but also to teach them as well to my siblings and class fellows . YOU are a BLESSING to all your STUDENTS around the world !!! ALWAYS be Blessed , Happy and Healthy . ... Thank you very very very much ...

[@MathswithMuneer]

He literally inspired many, that is why young teachers like me are also in this profession.

[@ricardocorrea9398]

I took precal and prereqs all in high school and never understood what f(x) meant. I just knew it was interchangeable with y. Thank you professor Leonard. Excited to pass calculus with your guidance.

[@matzr3000]

I used to hate life, but now im excited abt slopes of a parabola

[@bhargav7476]

I'm so happy that I picked this over Khan Academy Precalculus

[@nathanmgg]

God bless this man. Must protect at all costs.

[@janjan-yn1eb]

saw my co worker doing calculus 2. got intrigued. mind you, i don't have any mathematical background aside from high school algebra, my teacher for trigo was so bad at explaining and i never got to learn anything. tried jumping directly to calculus and couldn't even grasp the basics of it. im on your third video, the moment you started the graphs i got lost but eventually you managed to pull me back in. with just that last part discussion, i think i just might have a chance to actually learn calc. so thanks :)

[@Sir_Infiden]

I just realised from one comment that it's teachers day in India today and i am Indian too, so i really have wish you happy teacher's day sir and also, as an 8th grader, calculus was always hard for me to understand but your videos are the ones thst are now finally letting me to get a pace and be able to do some basic calculus

[@GarryBurgess]

Wow, I've seen this stuff several times before, but I loved the intro to this video, explaining better why the 2 notations for function ,and why f(x) is superior.

[@loganwillans4035]

That was the best explanation of function notation that I've ever heard! Bravo.

[@novasg3613]

This was very helpful. Math is easy to solve, but the hard part is knowing what equation to use and why so you understand what it’s asking of you. You show both that and how to solve. 🙏

[@ricks2907]

I think your an American hero. Will you be doing trigonometry. Long time supporter on fixed income that loves mathematics. Thanks for all you do. Grace and mercy to you my brother.

[@ProfessorLeonard]

Trig is a big yes!

[@hydromatic2688]

“whatever the f- the book tells you to do” 😂😂

[@josueabarca1913]

HAHAHAH i heard that too lmao I was looking for this comment

[@hjkjy4602]

This guy is really good at teaching this helped me out a lot. Thank you so much for this

[@yeetya3203]

Reviewing for my calc class and I got to say Professor Leonard is Goated. Finna write a write a 5 star rate my professor review lol The Goat Goat man Math Goat

[@RegularEverydayNormalGuy]

I really appreciate the approach that you took with this, explaining even the small things, it might make the video longer but it helps SO MUCH with filling the gaps in my math knowledge. Thank you so much

[@notomatoesx]

Please continue Differentials 😬; I’m taking both differentials and cal 3 and your notes are tremendous for cal 3. Thanks for everything you do for us students !!

[@ProfessorLeonard]

I will be :)

[@notomatoesx]

@@ProfessorLeonard you're the best, thanks for making life easier .

[@Cloud_Strife27]

These videos are all so great & appreciated !

[@dadisochitengwa1280]

I finally understand the difference quotient. Thank you very much.

[@maxallanfotchin1995]

Man teacher out her can't even be compare of your teaching you are so good at this

[@victoriatapia6449]

professor Leonard is a comedian. love the content

[@bobhope1160]

47:01 Explains why Difference Quotients are important (spoiler: Allows calculating slope of a curve based on point)

[@georgesadler7830]

Professor Leonard, thank you for a fine analysis and evaluation of functions and their impact on advanced mathematics. These concepts are the building blocks for a solid mathematics foundation.

[@DanaNourie]

Thank you sooooo much for all the explanations! I have to understand why in order to remember, and that’s something most math teachers don’t bother with. I need to understand context. I love the Difference Quotient preview and explanation. I tried learning before but was clueless as to what it was or why I always got the wrong answer. This is why I wanted Precalculus with Professor Leonard.

[@serafin8541]

Simply brilliant explanation

[@casatkinson]

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

[@stacygilbert3853]

facts

[@rga8895]

You're a great teacher, thank you for your videos

[@kes9716]

Wanting to cry the night before my first test. proff leonard is somewhat easing my pains

[@WaBC9IPa]

Wow I can't believe I finally understand this formula... This is the exact definition of derivative! You literally got the derivative by calculating this equation. Why couldn't I understood this in university?

[@babytime1]

cause its a debt trap, instead of an experience.

[@drvenomhd]

which criminal disliked the video?

[@funmaster5249]

somebody must have made a mistake

[@Peter_1986]

The person who disliked this video made a sign error.

[@skillsetsociety]

Watching Professor Leonard playing with maths>>>>> Netflix

[@EKDupre]

These videos enriched my life. Thank you Professor.

[@qariel8103]

I just had one question: (2x^2+1)/(x^2-5), he said you can't cancel anything or do anything more... I was confused when you can simplify and when you can't, because if it was multiplication instead of division I'd start distributing stuff to go further. Obviously I'm rusty on basic algebra so I'll have to learn more about this down the road. Absolutely phenomenal lecture once again, I'm steadily making my way through this playlist :)

[@Soratiwa288]

Thank you so much, I'm the fans from Thailand, Your explanation is really good and I can understand even before, I think I would be dead for Math. Thank you for making videos for us. Thank you so much !!!!!

[@MA-rc2eo]

Sir I'm one of your remote students and I thank you for making this video.

[@fuzyfuzfuz2]

"Okay pause and try on your own," (Writes down answer, feeling pretty good! Unpause video). "PLEASE don't give me x^2 +1." (Looks at answer with x^2 + 1)....... whoops 😂😂😂 thanks for helping me correct my mistakes!

[@anthonypeak1]

These lessons are topnotch!

[@callous21]

Definitely see now I missed a lot of stuff when I took precal. It's the main reason I struggled with calc 1 I believe. Well hopefully I'll be better prepared after studying this series of videos

[@thealdotrend1137]

Thank You! I Finally got the Different Quotient!

[@kingfuzzy2]

You are always such a wholesome and forward teaching per person thank you ( :

[@iyalodeolajubu6653]

at 16:48 where did you get 2x from

[@busrraengin]

Thank you for the explanation, Professor Leonard; it was super helpful.

[@ektabansal5332]

Professor really made me understand why something divide by 0 is undefined

[@indianarchangel]

Such an outstanding teacher. Mad respect

[@jclopez99]

Awesome teacher! Wish I'd had you. I would have fallen in love with math. Keep up the great work!

[@jasmingarcia5257]

Professor Leonard exudes king energy!

[@Squash101]

46:00 I'm watching this after Calc one an just watching to get ready for calc 2 an it feels like a movie that

[@wackzingo]

Function names don't have to be limited to g(x) or f(x) or even h(x) but you could even use words. It's definitely not common in math but there is no reason you couldn't. It is common in programming and you see that in graphing calculators. Ever seen a function named sqrt() instead of f(x)? You could do the same thing and write sqrt(x) = √x although I can't type the full symbol on my phone but you get the idea. Now to really blow your mind, there is no reason you couldn't use a symbol like an emoji for the function name. You could do 😀(x) = x*x to represent a function that squares a number.

[@theoendre6484]

Thank you professor for the cool lecture

[@selmar7991]

I just found you and I love you 🤣 that taylor swift joke got me 🤣🤣

[@anupammishra2719]

Happy Teachers day professor 😄😄

[@kyle7362]

I wish my teachers/profs were half as good as you at teaching

[@thatblackcbr04]

I love this guy so much !!!

[@walthexwottrich9774]

in minute 49:58 , that part of the slope 1 to side and 4 up, I didn't understand, that 1 of the side, were did it came from?

[@pepkibirb319]

IT FINALLY MAKES SENSE

[@chemicalengineeringbd7614]

Thank you legendary prodessor leonard🙏💓

[@kianab8649]

It's the Taylor Swift "Blank Space" reference for me 😂 Thank you for making math clear and humorous

[@sabrinae.6361]

I didn’t come for the math but stayed 😂