❖ Composition of Functions ❖
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❖ Composition of Functions ❖

August 25, 2019


In this video we’re going to talk about composition of functions And just the notation, they’ll write F, circle, G of X This is not a multiplication symbol. That’s why they try it like a little dot. You know denotes. That’s why they try to make it a little bit bigger Likewise, it’s not the word fog I’ve heard people ask me about fog x and I was never quite sure what they were talking about so It’s read F of G of x, and what this notation means is you’re basically plugging g of x inside of f of x likewise g of f of x means take f of x and plug it inside of G So I think composition is something that [confuses] people a little bit, so let’s take a couple examples so f of x let’s make that x squared plus x and Try to color code things here to make them a little bit easier to follow suppose we have g of x is let’s make [it] 4 minus x all right and A Couple things here suppose. We want to figure out f of g of x [so] again what that means is we want f of g of x and Well, I can plug in g of x I know what g of x is all right g of x is the equation [4] minus x So that’s what I’m going to substitute in there okay, so there’s [f] of g of x and you know kind of off to the side if somebody said figure out f of 3 what would you do well you would look back at your original f of x equation and everywhere there was an x you would replace it with a 3 So I’m going to do the same thing when I do this problem I’m going to go back and everywhere. There’s an x I have to put something in But this time the thing that I’m substituting in is not just [a] simple number like 3 it says I have to substitute in the quantity quantity excuse me 4 minus x so this is the idea with a composition of functions Instead of just kind of substituting a simple number in you’re putting in a more complicated expression So let’s simplify this one down. This is 4 minus x times 4 minus x plus just 4 minus x if I foil this out. I’ll get 4 times 4 which is 16 I’ll get 4 times negative x which is negative 4x negative x times 4 that’s negative 4x Negative x times negative x that’s positive x squared I have my plus 4 I’m running out of room here and my minus x still hanging out and now it’s just a matter of combining like terms, and I usually think take things in descending power, so I see an x squared I’ll write him first a Negative 4x and a negative 4x that’s negative 8x Minus another x that’s negative 9x so that takes care [of] my x terms And then I’ve got 16 plus 4 which is 20 and that would be the composition f of g of x Okay, let’s maybe take a variation on this one and let’s flip it around So again, I had that f of x is x squared Plus x and I had g of x is 4 minus x suppose now I want to find g of f of x and recall multiplication the order in which you do things doesn’t matter 2 times 3 is the same as 3 times 2 and This notation again. It almost looks like Multiplication, but it’s not so you have to be careful because in general you don’t get the same thing So notice I found f of g [of] x to be this stuff Now let’s see what we get when we do g of f of x so [again] this notation it says you plug f of x into [your] G function well again f of x is The [function] x squared plus 4 excuse me x squared plus x so it says I’m going to plug in x squared Plus x and Just like before if it said find g of 2 I would [find] my G function everywhere there was an x I would plug in a 2 Well now it says everywhere. There’s an x I Have to plug in all of this mess x squared plus x and this is now going to be the composition function g of f of x so if I distribute I’ll get negative x squared [I’ll] get a minus x term and then I still have my positive 4 hanging out and that’ll be the composition, so Notice I got something much different This is what I got when I did f of g of x and this is what I got when I did g of f of x so two totally different things there so just something to be aware of Suppose the original problem was to find g of f of [four] Let’s make it three just so I have a different number floating around the first thing I would do just like before [I] would just find g of f of x just like it is and I would get this stuff And now it says all you’re doing is you’re [just] replacing the x’s with [3S] So the [x’s] with threes, so that’s what I would do here and we get 3 squared minus 3 Plus 4 well, you square the 3 first so I’ll get 9 but then it will become negative 9 I’ve got a minus 3 hanging out a positive 4 well negative 9 [minus] 3 that’s negative 12 plus 4 that’ll turn out to be negative 8 whoops is all cut off negative 8 and that’ll be my solution So this is the basic idea with composition of functions You’re basically just putting one function inside of the other one So I hope these couple examples make some sense I definitely encourage [you] to try to rework them if they’re don’t if they don’t make a lot of sense Certainly feel free to send me an email. You can visit my website Just math tutoring [comm] there’s lots of other [examples] on there if you need to see some more Composition of function examples send me an email let me know what your problems are and I’ll try to get to them as soon as I can and I can post another video with a few more a few [more] examples on there, so Good luck with this stuff I think composition is probably of all the kind of operations with functions it can be the most confusing so Just try a few examples, and I think you’ll get there

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  1. OMG, I was so stuck until I found your video. It made it easier because of the color coordination I think whatever the case was, I completed all of my questions and got them correct in a manner of minutes versus days prior to finding your video. THANK YOU. Keep up the great work

  2. If I would become a billionaire someday. I would donate you half of my net worth just because you teach what "formal teachers" fail to teach effectively.

  3. wait, in the last example, you did the x^2 first then applied the negative, but isn't that wrong? isn't it (-x)^2, which would result to x^2?

  4. I learned composition in just 8 minutes rather than my teacher teaching 1 and half an hour thank you for sharing this bro

  5. Wow!Thanks a lot…I happen to be a slow learner..your patience made me understand this …I re watched 3 times to perfect it..hoping to tackle my exam this Friday well..

  6. Thank you i was so confused somewhere between the sum and finally you and youtube helped me go on hit 1M sooooon

  7. thank you bt this are so simply i have triky peoblems.
    when 1 function is given and the solution of the orther functions are given then you are ask to fine fog…..

  8. Please more examples. Im getting a 72 in my class and we are doing these now. My teacher missed a class and is cramming this in a hurry because he took that day off. One more test then the final. Im totally having difficulty. I cant get below a 70 in this class. Thank you! You are calming and understand that every little thing can help like using different colors and explaining ways people tend to get confused

  9. Thanks, was able to complete my section on this on Aleks. could not have done it with out watching this video. Thanks again.

  10. Hey Patrick, I have a question. My math final is on Monday and she barely introduced this concept. On the first example of composition of a function, the final answer was x^2-9x+20. Would you factor it or leave it in that standard form?

  11. Also doing this method seems very complicated. I have another question, if you (g o f) (x) where did the 3 come from? I've watched this video multiple times, I just dont quite understand.

  12. My professor yelled at me and told me she wants to murder me because she skipped a bunch of steps you showed and I didn’t get it. This is why I hate math, all the step skipping, and lack of coherent written instructions! Thank you!

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