Log in Hannah 11 years agoPosted 11 years ago. Direct link to Hannah's post “at 7:26 why is y a square...” at • (173 votes) Brendan 11 years agoPosted 11 years ago. Direct link to Brendan's post “You have to remember that...” You have to remember that in algebra, what is done to one side of the equation has to also be done to the other side of the equation. When y^2 = 3, in order to find out what y is equal to, you have to get rid of the square. If you square 3, you also have to square the other side of the equation to make it equal. 3^2 would be equal to y^4, which doesn't really help us. Instead, get rid of the square by getting the square root of y squared (which is equal to y) and then finding the square root of the 3. (424 votes) faz919 11 years agoPosted 11 years ago. Direct link to faz919's post “So, one day, I asked my d...” So, one day, I asked my dad if a function could be graphed as a circle. He said yes. But I said no because I thought there would be more than one output of the input. For example, see this program I made: • (176 votes) Chuck Towle 11 years agoPosted 11 years ago. Direct link to Chuck Towle's post “Ωπ fαz919 πΩ ,By defini...” Ωπ fαz919 πΩ , By definition of a function, a circle cannot be a solution to a function. But a circle can be graphed by two functions on the same graph. If you look at this program, you will see that it used two functions to create the graph: http://www.khanacademy.org/cs/xr-yr-1/1807411349 (204 votes) WillMosher 11 years agoPosted 11 years ago. Direct link to WillMosher's post “What's the difference bet...” What's the difference between functions in algebra and functions in programming languages? Is here one? • (126 votes) jimstanley49 11 years agoPosted 11 years ago. Direct link to jimstanley49's post “They are very similar. I ...” They are very similar. I think the best way to put it is that a math function takes in some mathematical construct (be it a number or some other variable or even another function) and spits out a mathematical construct, usually a value. (44 votes) ilovemeat123 11 years agoPosted 11 years ago. Direct link to ilovemeat123's post “I don't get how Sal got h...” I don't get how Sal got h(2)=3 and h(8)=11. someone help please? • (78 votes) Nitin Rajput 5 years agoPosted 5 years ago. Direct link to Nitin Rajput's post “_The next largest number ...” The next largest number that starts with the same letter as 2 is 3. (39 votes) raqinda 11 years agoPosted 11 years ago. Direct link to raqinda's post “why are there two variabl...” why are there two variables? • (20 votes) David Solorzano 11 years agoPosted 11 years ago. Direct link to David Solorzano's post “If you mean, "why does f(...” If you mean, "why does f(x) contain two variables?", please note the f is not a variable. The f is just a way for you to know that when you see f(x) to treat it as a function and not mistakenly treat it as multiplying one variable by the other (it DOES NOT mean f multiplied by x). It does not have to be an f, it can be any symbol and using different symbols such as h(a) helps differentiate one function from another. (59 votes) tarfail 6 years agoPosted 6 years ago. Direct link to tarfail's post “Can a function have multi...” Can a function have multiple inputs? If so, how would you graph said function? • (13 votes) Kim Seidel 6 years agoPosted 6 years ago. Direct link to Kim Seidel's post “Yes, a function can have ...” Yes, a function can have multiple inputs. We can graph in the coordinate plane when we have 1 input to 1 output. If we have a function with 2 inputs to create 1 output, we can graph in a 3 dimensional graph of (x, y, z). Once you go to even higher inputs, we typically would not graph them as we don't what a 4-dimensional space looks like. (25 votes) Toska Pi 5 years agoPosted 5 years ago. Direct link to Toska Pi's post “Hi! I was wondering if th...” Hi! I was wondering if there is a relationship between the equation describing how to plot a circle ( • (10 votes) itimespi 5 years agoPosted 5 years ago. Direct link to itimespi's post “Yep, there definitely is....” Yep, there definitely is. Let's consider a circle with center (0, 0) (to make the explanation a little simpler) and radius 3. Let's find some points on the outer edge of the circle. A noticeable one is (3, 0) (3 units away from the center). Let's try to make a right triangle, where the center of the circle is one vertex, and its opposite vertex is the outer edge. Since this is a right triangle, we should be able to apply the Pythagorean theorem. The base of the triangle would be the x-axis, and the adjacent side would be some y-value. The hypotenuse would be the radius of our circle. Thus, a = x, b = y, and c = r. Using this in the Pythagorean theorem, we find: Does this work for the point we selected (aka (3, 0))? You will find that this works for every single point on the circle. For example, another point on our circle is (3/√2, 3/√2). Does this work in our equation? When you use this equation with every possible x-value and y-value and graph the points you are able to make, you will construct a circle. (23 votes) Jonathan Ziesmer 10 years agoPosted 10 years ago. Direct link to Jonathan Ziesmer's post “What does the f stand for...” What does the f stand for? • (7 votes) Eli Rochelle 10 years agoPosted 10 years ago. Direct link to Eli Rochelle's post “Basically, the concept of...” Basically, the concept of functions gives us a way to name the whole For example, the first function you showed can be called 'squaring', We can also treat these names like variables, where we don't know what (10 votes) timesplus 3 months agoPosted 3 months ago. Direct link to timesplus's post “I want to ask, is it a fu...” I want to ask, is it a function that is important? What I mean is that even if an equation does not satisfy the definition of a function, we can still get its results. Is there any special reason why we define a function • (7 votes) TheReal3A 3 months agoPosted 3 months ago. Direct link to TheReal3A's post “Oof, it's hard... I'll tr...” Oof, it's hard... I'll try explain this as best as i can. They have different purposes. Equations encompass anything that has to equal another thing, and are most useful to solve for variables. Functions are a type of relation and are most useful to analyse quantities and be graphed. The concept of functions mapping input onto (at most) If an equation has exactly two variables involved in it, there is likely a relation between those variables, which is implied by the equality of both expressions. Rearranging to solve for the variable you want to measure, you get a function (as long as there's not multiple outputs for any input)! Functions involve anything with an independent and dependent variable. Height -> Volume, Time -> Temperature and Sales -> Profit are all examples. (16 votes) X Y 7 years agoPosted 7 years ago. Direct link to X Y's post “How can we graph the fanc...” How can we graph the fancy function h(a)? • (7 votes) 🍕⚡ ViςhαL Πaudel⚡🍕 7 years agoPosted 7 years ago. Direct link to 🍕⚡ ViςhαL Πaudel⚡🍕's post “Good question, but, we ne...” Good question, but, we need more info! For essence 'h (a) = x*a + b', where 'x' is the slope of the function and 'b' is the Y-axis intersection, or what you call is as a "Y intersect". ( Remember 'b' can either be positive or negative) A function is nothing but a number "operator" it takes some numbers & variables, fiddles with it and gives only one output. Thanks! May you have a great Day/Evening ;] . Extra: Well, if you are given an expression for h(a) which is not necessarily linear(that is not of the form of ax+b) then you can graph it computing various output points on various inputs and get a sense of the function's graph(you can use some concepts of how the function progresses or looks like using calculus and understand the end behaviour). (14 votes)Want to join the conversation?
7:26
https://www.khanacademy.org/cs/a-function/1860626452
My dad said yes because he said you could find the absolute value of both sides, but I didn't think of it that way. Can someone tell me: who is correct?
A function, by definition, can only have one output value for any input value. So this is one of the few times your Dad may be incorrect. A circle can be defined by an equation, but the equation is not a function.
y=√(r²-x²) and y=-√(r²-x²)
In programming, a function takes in some construct that is defined by the programming language (numbers, strings, classes, the results of another function) and returns a construct defined by the language.
They both start with the letter t. In the same way, The next largest number that starts with the same letter as 8 is 11. They both start with e. 6:30
x² + y² = r²
3² + 0² = 3² → 9 + 0 = 9 → 9 = 9 ✓
(3/√2)² + (3/√2)² = 3² → (9/2) + (9/2) = 9 → 18/2 = 9 → 9 = 9 ✓
process of evaluating a particular expression, so we can talk about
it as a whole. We can compare different functions, discuss their
properties, or actually operate on functions to make new functions.
It also broadens the concept, because not all functions can be
written as a simple expression. These two processes, naming things
and extending them, are central to what mathematics is all about.
and the second can be called 'adding 3'; but most functions would
have to have much more complicated names. By calling one F and the
other G, we have a simple way to discuss them. Some functions, like
the square root and the absolute value, can't be expressed in terms
of more basic functions, but only by inventing a whole new symbol. In
fact, we like to write the square root as 'sqrt(x)', using function
notation, because we don't have the symbol available in e-mail.
specific functions we are calling f and g, yet we can say general
things about the relation of f and g, proving that something is true
for ANY functions, or at least for any functions of a certain type,
all at once. That is powerful!
Describing something as a function is sort of labelling it.
Equations show the equality of two expressions.
Functions are more specific than equations; they show a singular relationship between what is changed and what is measured.1 output separates functions from equations.
Note that many formulae or conversions can also be described as functions, such as Circle Radius -> Circle Area, Fahrenheit -> Celsius, Degrees -> Radians, and so on...
Video transcript
A function-- and I'mgoing to speak about it in very abstract terms rightnow-- is something that will take an input, andit'll munch on that input, it'll look at that input,it will do something to that input. And based on what that input is,it will produce a given output. What is an exampleof a function? I could have somethinglike f of x-- and x tends to be the variablemost used for an input into the function. And the name of afunction, f tends to be the most-used variable. But we'll see that you canuse others-- is equal to, let's say, x squared,if x is even. And let's say it is equalto x plus 5, if x is odd. What would happen if weinput 2 into this function? The way that we woulddenote inputting 2 is that we would wantto evaluate f of 2. This is saying, let's input2 into our function f. And everywhere wesee this x here, this variable-- you can kindof use as a placeholder-- let's replace it with our input. So let's see. If 2 is even, do 2 squared. If 2 is odd, do 2 plus 5. Well, 2 is even, so we'regoing to do 2 squared. In this case, f of 2 isgoing to be 2 squared, or 4. Now what would f of 3 be? Well, once again, everywherewe see this variable, we'll replace it with our input. So f of 3, 3 squared if 3 iseven, 3 plus 5 if 3 is odd. Well, 3 is odd, so it'sgoing to be 3 plus 5. It is going to be equal to 8. You might say, OK,that's neat, Sal. This was kind ofan interesting way to define a function, a way tokind of munch on these numbers. But I could have done thiswith traditional equations in some way, especiallyif you allowed me to use the squirrellybracket thing. What can a function do thatmaybe my traditional toolkits might have not beenas expressive about? Well, you could even doa function like this. Let me not use fand x anymore, just to show you that the notationis more general than that. I could say h of a is equal tothe next largest number that starts with the sameletter as variable a. And we're going to assumethat we're dealing in English. Given that, what ish of 2 going to be? Well, 2 starts with a T.What's the next largest number that starts with a T? Well, it's goingto be equal to 3. Now what would h of-- I don'tknow, let's think about this, h of 8 be equal to? Well, 8 starts with an E.The next largest number that starts with an E-- it'snot 9, 10-- it would be 11. And so now you see it's avery, very, very general tool. This h function that we justdefined, we'll look at it. We'll look at theletter that the number starts with in English. So it's doing this really,really, really, really wacky thing. Now not all functionshave to be this wacky. In fact, you have alreadybeen dealing with functions. You have seen things likey is equal to x plus 1. This can be viewedas a function. We could write thisas y is a function of x, which isequal to x plus 1. If you give as aninput-- let me write it this way-- for example,when x is 0 we could say f of 0 is equal to, well, you take 0. You add 1. It's equal to 1. f of 2 is equal to 2. You've already done this before. You've done thingswhere you said, look, let me make a table ofx and put our y's there. When x is 0, y is 1. I'm sorry. I made a little mistake. Where f of 2 is equal to 3. And you've done this beforewith tables where you say, look, x and y. When x is 0, y is 1. When x is 2, y is 3. You might say, well, whatwas the whole point of using the functionnotation here to say f of x is equal to x plus 1? The whole point is to thinkin these more general terms. For something likethis, you didn't really have to introducefunction notations. But it doesn't hurt to introducefunction notations because it makes it very clear thatthe function takes an input, takes my x-- in thisdefinition it munches on it. It says, OK, x plus 1. And then it produces1 more than it. So here, whatever the inputis, the output is 1 more than that original function. Now I know what you're asking. All right. Well, what is nota function then? Well, remember,we said a function is something that takes aninput and produces only one possible output forthat given input. For example-- and letme look at a visual way of thinking about a functionthis time, or a relationship, I should say-- let'ssay that's our y-axis, and this right overhere is our x-axis. Let me draw a circlehere that has radius 2. So it's a circle of radius 2. This is negative 2. This is positive 2. This is negative 2. So my circle, it'scentered at the origin. It has radius 2. That's my best attemptat drawing the circle. Let me fill it in. So this is a circle. The equation ofthis circle is going to be x squared plus y squaredis equal to the radius squared, is equal to 2 squared,or it's equal to 4. The question is, is thisrelationship between x and y-- here I've expressedit as an equation. Here I've visually drawn allof the x's and y's that satisfy this equation-- is thisrelationship between x and y a function? And we can seevisually that it's not going to be a function. You pick a given x. Let's say x is equal to 1. There's two possible y'sthat are associated with it, this y up here andthis y down here. We could even solve for thatby looking at the equation. When x is equal to 1, weget 1 squared plus y squared is equal to 4. 1 plus y squared is equal to 4. Or subtracting 1 from bothsides, y squared is equal to 3. Or y is equal to the positive orthe negative square root of 3. This right over here is thepositive square root of 3, and this right over here isthe negative square root of 3. So this situation,this relationship where I inputted a 1into my little box here, and associated withthe 1, I associate both a positive square root of3 and a negative square root of 3, this is not a function. I cannot associate with myinput two different outputs. I can only have oneoutput for a given input.
