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Understanding Functions
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0:00
okay let's see if we can understand functions in about seven minutes and that's always my target uh goal with my videos here is to try to teach you something in about seven minutes sometimes uh very often i go a little bit over but in a short period of time if you stick with me you're going to learn a lot about functions if you're struggling in this topic don't feel bad a lot of students struggle with functions it's a huge topic in mathematics and it's extremely important that you
0:29
understand it so really the main goal for this video is just to get you to strengthen your fundamental understanding of what a function is so that's kind of the objective here but before we get going let me go ahead and introduce myself my name is john i'm the founder of taba class math i'm also a middle and high school math teacher and i have many online math courses i teach fun functions very in-depth in several of my courses but i'm going to suggest that you check out my algebra course
0:58
really get into functions heavily there i'm going to leave the link to that in the description of this video if you want to learn more really master functions but let's get into functions now so here's an example of what a function looks like so there's all types of functions again it's a huge topic in mathematics but when you see a little notation like this that indicates some sort of function now this the way we state this or read this is f of x okay this is the name of this function
1:29
and any function like so f of x is equal to the variable y all right so you can think of this as the function three x squared plus 1 but it's equivalent to the equation 3x squared plus 1. okay so it's a little technical difference there but equations okay like this is a quadratic equation and we would refer to this as a quadratic function okay again we say this is uh as f of x is equal to three x squared plus one but we could have all kinds of functions for example we could have something like
2:07
this that would be g of x that's the way we would state that if i had this h of t i would write it that way so again in mathematics all sorts of functions but this is a real basic example to kind of stress what i want to teach you here okay now what is a function um well a function is basically some sort of rule okay a rule in the sense that we're going to input values into the function all right and then we have some sort of rule okay now the rule basically says okay take your input values
2:47
the rule says do something with that value like manipulate it in some way okay and when we're done with that we're going to get some sort of output value okay now the input value inside into a function we're going to call this the domain all right so this word here so all the numbers that we can put into a function is called the domain and then the respective output values is called the range okay so really important words again [Music] depending on where you're at with
3:21
functions you're going to study the domain and range at a much more advanced level again functions are a huge topic but let's just make sure you get the the basics down uh in this video okay so let me go and erase this all right so we have the input which is the domain well the set of values that you're allowed to put into a function it's called the domain okay it's going to be associated with the x variable let's go ahead and and establish that and then the output is going to be a step
3:50
associated with the y variable right you'll see that here in a second what i'm talking about let's look at an example of how to evaluate a function so let's go ahead and figure out what this function's value is at two so that means i want to evaluate that's how we would state that this function we want to find f of two okay so here again is the input so you can see i'm replacing this x with the two so over here in the rule part i'm going to replace this x with whatever value i plugged in for
4:22
this x here okay of course this is going to be 2 squared plus 1. so now at this point i can go ahead and evaluate or finish evaluating this function by just being very careful you know all of you out there are experts in order of operations so we're going to be careful here because those are the common mistakes common mistake where students um make errors is uh they'll go oh 3 times 2 that's 6 and they square that remember you got to be very careful with the order of operations
4:51
this is going to be 2 squared so that's going to be 4 3 times 4 plus 1 which of course is going to be 12 plus 1 or 13. okay so we have f of 2 was equal to 13. now again let's go ahead and take a look at this function in a different way i want to really stress something here so when we had the x we had x equal to 2 our y value is 13. now why is it the y value remember i said y is the same thing as f of x so our f of x our output was 13 but you want to associate that with y okay so whatever your
5:32
y value here let's just say y was 13 one x was two because you can construct a table like so 213 but this corresponds to the ordered pair or point on the x y axis to 13. that's a point okay let's do it over here okay so i have an x i have a y let me erase this so the point 2 13 1 2 let's say it's up here 213 is on this function here okay graphically speaking so we can look at functions in a variety of different ways we can look at them at uh in terms of their table okay how they're
6:17
represented or sets of points or algebraically by their actual function rule so this particular function happens to be a parabola so it's going to start at 1. it's going to look something like so okay and all the points that are on this graph all right are part of the domain and range okay now i don't want to go ahead and start taking this um topic you know beyond seven minutes i'm probably 47 beyond seven minutes right now because again functions are a huge huge part of mathematics but
6:56
before i uh end the video let's talk about some other follow-on things that you're going to be um learning with respect to functions okay and each one of these are really important so first of all you just have to understand what a function is okay and oftentimes functions let's just do this real quick okay we have something in math called relations okay and a subset of relations are functions okay so all functions are relations but not all relations are functions okay so one of the things
7:28
that you need to understand about functions is uh knowing how to determine when you're dealing with the function or uh when some if if something is a function or let me state it that way that's a big part of what you should be learning in your math course another thing is function operations so if i have two functions let's say i have f of x is something and i have g of x is something else well you can manipulate functions you can add them you can multiply them you can do all kinds of stuff
8:00
you can do what we call a composite function all right so f of our g of f of x can do all sorts of stuff you can also find function inverse all right which is going to be another huge topic so again many many facets of functions that's why um you know it doesn't make sense for me to try to squeeze this in in seven minutes that would just be uh i don't even know if i could do it but what i wanted to do was to stress the fundamentals of what a function is and even you know
8:30
i could have even taken more of my time to do that but that's going to wrap up this video hopefully this kind of um was a good little refresher for you again uh at this time i've been on youtube for a good 12 years i have literally hundreds and hundreds of math videos posting all the time so hopefully you consider subscribing if you like this video definitely appreciate a thumbs up and leave me some feedback um obviously you're interested in functions what math course
8:56
are you taking algebra algebra 2 college algebra it's taught at many levels even in calculus pre-calculus excuse me so any feedback is good feedback and i'm going to leave a link to my algebra course in the description of this video now i teach functions thoroughly in my all my algebra courses so um if you're in another course like algebra 2 or something different you can uh check out my course catalog and you can kind of go to that particular course but if you want to know you know everything
9:31
i was talking about with functions you can definitely find it in my algebra course but with that being said i definitely wish you all the best in your mathematics journey and have a great day
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