calculus studies functions and this

course is no exception, we’re going to be studying functions in this course, alright…

functions like f(x) or the sine function or the square root function and that raises a question… what are

functions? if we’re gonna be studying functions we should know what they are that question is pretty metaphysical i mean what are numbers? what’s the number four? i mean, that question doesn’t make a lot of

sense does it.. i know it’s not like the number four is a physical object that i

can go look at it and see if it’s got polka dots or stripes or something, right? but i know four objects when i see them. I sort

of understand what numbers do more than what they are

in some metaphysical sense the same is true about functions I’m not really going to tell you what a function is what am gonna tell you is what a function

does this is what a function does a function of size each number in it’s domain, another number this definition doesn’t say anything about how the

function does that assignment let’s see an example to make up some example suppose i’ve got some function, I’ll call it ‘f’

f for function maybe this function assigns to the

number two the number four so alright f of two is four or f of three is nine or f of four is sixteen or f of five is twenty five I’m just making this up there’s some function and telling you

what number it assigns to each number f assigns the number two the

number four f of three is nine right so at the signs to the number three the

number nine other than that is gonna listen off every single assignment that ethnics so instead one way to talk about these

assignments is to use a rule like f_ of x_ equals xc where and the single rule explains how all of

these assignments are made this rule says that af assigns to the

number x the number x square so in particular epa science the number five five square twenty fiber emphasize the number four

four-square in which a sixteen yr appa sized the number three three squared which is not so a lot of times when you actually talk

about how these assignments are being made you some sort of rule like this new

right path of acts is something to compute the output value so function assigns to teach number in

his domain another number one way to do that before the rule course this definition of function

involves another word domain what is the domain a muscle as they otherwise the domain

consists of all numbers for which the rule makes sense a function assigns each number in this

domain another number and the domain is just the numbers that

i can plug in let’s take a look to see what i mean by

this spose i’ve got a function af of acts

equals one over x so it’s a function given by rule it takes an input x introduces the

output one overexpressed besides the number x the number one oh for acts but this rule doesn’t always makes sense

right and dividing by acts hi and uh… to divide by acts what do i need i must have that x is nazirah not permitted to divide by zero so i can plugin a number for x except

for zero and that’s when this will make sense soul summarize that by saying the domain of af is all real numbers expect sole function take some important produces an output

that’s what it does but how is it supposed to make us feel houses to imagine that who’s one metaphor that she could use to

try to think about what a functioning is actually doing you could imagine our conveyor belt with the function you know when you can actually numbers

coming in boom being hit by the functioning going out transform somehow

by whatever the rule is that the function is what i’m talking about got a conveyor

belt here got this big box a management that big box is the

function in here i’ve got the input i met in the number five has to be with us machine is gonna do

maybe this is the function af of etc sequels xc where’d plus x and i’ve got this number five here also

the conveyor belt going to the number five starts moving through the machine and win the machine is done processing

unit comes out from the other side and now with the number thirty but because f of five is five squared plus five which is twenty five plus five start so we’ve seen how you can write down a

function by using up our rule involving these these mathematical symbol protect

squared plus tax you can also write out a function just

using english words so let’s see an example of that make up a new function

here or call a part of acts well-defined our backs to be serai sf and it’s three times thrice the square

or have x u_s_ computed some some values of the function a like the function et

four is six allies that well are four would be three times the square root of

four which is three times to which is six alcatel course i do the calculation this way you

know it’s not too surprising that instead of writing a

sound words thrice the square root of x and that could have just written three times the square taxed so i’ll

give you not to impress to this the point is that you can define a function just by

writing down with the function is supposed to do using english words so we’ve seen an example of a function

that we define just in terms of algebra apple vaccine quebec squared plus x vogelstein example of a function that i

define entirely would just english words bringing the combine those two things

right we can use the english to sort of picked out different kinds of uh… of

algebra busing example that now so there’s another function is made up achieve acts is x square if taxes bigger than or

equal to five and twenty six if texas less than five the point here is that i’m using just a

little bit of english this word d_f_ in order to select based on how big axes by different elgin brea ca rule for

calculating the function as is alleged check let me just do some

calculations and uh… that might convince you know

how how this notations so-called piecewise notation works so what is g of one well let’s take a look one is less than

five so that means that he’s the second rule

for calculating g soc two times one wishes to you was g_s_a_ four before still less than five so he’s the second rule again twice the input two times for nancy

outputs eight but what’s a g of five five is not less than five five is

critical of five so this if is telling me to use the

first of the two rules for calculating g seminar used five squared as the output i’ll be twenty five or uh… g of ten wiltel is definitely bigger than or

equal to five times bigger than five so i can use the first rule and the

alpert to this function is computed is ten square or one hundred so this is gonna really more complicated

than just using some algebra right i’m using these if statements g select which of these two rules g will use to computers output uh… in principle mcfun’s whiskey

really complicated i mean all the samples from senior just

doing various kinds of algebra i mean eighty different algebra depending upon

which values of x and putting in but it’s all the now adding subtracting off

point about and that sort of thing if you can do really complicated things

with with functions let’s listen example of something that side of really

different than just doing a straight-out and a much crazier example at sea of

excuse the number of even digits in the number acts which x’s a whole number it’s zero if x isn’t a whole number so

otherwise with really different kind of crazy

function right socio say seven thousand children thirty-six is too why’s that this is all the amber so he’s the first

part of the rule and the number of even digits in this

number how i count the mets audit even aren’t even took two and six or at the two even

digits the value that functions too well let’s take a look at this number

here six two thousand two hundred to that again is a whole number so uh… the valley with the function at

sixty thousand troops who is the number of even digits imax and this member has how many even digits one two three four five the raleigh

vintage it so that values five here’s another example is another five

digit number two to three thousand five hundred

thirty one again it’s a whole number so he’s the first part of the rule the

number of even digits i count how many these digits are even five oscar know even digits of the value

of this function is euro at this point dot i plug in a number that’s not a

whole number like this this is not a whole number that i use the otherwise part of the

definition and the function at that point is just the right so we can use english to define a

function but gonna be careful i’d sometimes in retina

definition of a functioning might write down something that’s more ambiguous

than did you intended so let’s try define a function is making

this stuff up bahama function b effects for bad and also the value of the of axes some

rearrangement the digits of acts and it’s okay that i’m using english to

defy my function to something wrong with that but this definition is too ambiguous to

be the definition of a function here’s an example of what goes wrong was the functions value at three hundred

and fifty two well the function takes as input and

rearrange his the digits somehow team i think the functions value between

if you too yesterday turned twenty five because theater in twenty five a summary

arrangement of the digits of two hundred fifty two wrecked by two to five of the

two structure positions the the functions value its return to

the two should also be two hundred and

thirty-five is two three five is also a

rearrangement three five two the functions valley with three five two

should also be five thirty two because five three two was summary

arrangement double digits of three five two this is terrible a function disposes tickets input introduce unambiguously the single

output value but this so-called function takes this single input value in purportedly produces all these

different possible outputs this thing here is not a function a function takes one input and produces one output

too bad! i missed the deadline to sign up for the class…and you don't have the other videos posted to watch….ahhhh! won't you put 'em up on line? please!

Hi David,

Thanks for the comment. Even though the course is getting ready to start its fourth week, you can still sign up to access the course, lectures, exercises, and also enter the mooculus! Simply click the link in my video description above to sign up.