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Checking if an equation represents a function

Sal determines if y is a function of x from looking at an equation. Created by Sal Khan.

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  • piceratops ultimate style avatar for user Jay R. Ligon
    Is it safe to say that if there are exponents in the relationship that it will not be a function? Is this generally true? Thanks
    (4 votes)
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  • aqualine ultimate style avatar for user Andromeda
    but how did he make that arc? How will you know what shape to draw on a line?
    (2 votes)
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    • male robot hal style avatar for user jhenrypalmer
      Not sure yet dude, but one thing I've found on Khan Academy is to trust Sal and then later on you find the answers down the track. I've found that when he introduces concepts you don't understand, generally it seems you don't need to understand how to do them at this stage (if you are following through a logical progression). Looking at an equation and being able to draw it is probably something that comes from a ton of experience!
      (9 votes)
  • piceratops seedling style avatar for user Rain
    If you restricted the range to y>0 could you turn this into a function?
    (2 votes)
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  • leafers seedling style avatar for user Austin Watkins
    So from what I understand a functions can't be any value that's taken to an even power (but the rules and inputs can be). What other mathematical concepts are there that prevent a relation from being a function?
    (1 vote)
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  • mr pants teal style avatar for user Goel.Deepak9
    Couldn't I just plug this equation into my graphing calculator since they are allowed in most high school math classes and the math sats?
    (0 votes)
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    • piceratops ultimate style avatar for user Just Keith
      That depends on whether you understand the concepts. It is fine to have software to help you get through the tedious computations, but if you do not understand how to do the graph yourself, your calculator won't be much help.

      For example, you may be given a graph and asked what equation it represents. Your graphing calculator won't help you much there.
      (7 votes)
  • aqualine ultimate style avatar for user maddayst
    i have a friend who does not understand functions. he can't wrap his brain around it. what tips can help him understand?
    (2 votes)
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  • leaf yellow style avatar for user Mo
    Is there a section on graphing non-simplifiable radical expressions such as the one Sal did around the minute mark?
    (2 votes)
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  • male robot hal style avatar for user Ryan Domm
    At , how does Sal just know that y=square root of (x-3) gives off that curved line. Similarly, with the negative version... It's not even in proper slope-intercept form so how the heck does he know where the hell the line is going. Someone please help!
    (2 votes)
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  • male robot hal style avatar for user Ryan Domm
    At
    How do you suppose "y=square root of (x-3)" or the negative version gives you that curved line? That literally makes zero sense to me. He doesn't even have it in slope-intercept form to do that. Someone please explain!
    (2 votes)
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  • leaf green style avatar for user Ron
    if you squared any input does that automatically give you two potential outputs, because of the potential for positive and negative answers.
    (1 vote)
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Video transcript

In the relation x is equal to y squared plus 3, can y be represented as a mathematical function of x? So the way they've written it, x is being represented as a mathematical function of y. We could even say that x as a function of y is equal to y squared plus 3. Now, let's see if we can do it the other way around, if we can represent y as a function of x. So one way you could think about it is you could essentially try to solve for y here. So let's do that. So I have x is equal to y squared plus 3. Subtract 3 from both sides, you get x minus 3 is equal to y squared. Now, the next step is going to be tricky, x minus 3 is equal to y squared. So y could be equal to-- and I'm just going to swap the sides. y could be equal to-- if we take the square root of both sides, it could be the positive square root of x minus 3, or it could be the negative square root. Or y could be the negative square root of x minus 3. If you don't believe me, square both sides of this. You'll get y squared is equal to x minus 3. Square both sides of this, you're going to get y squared is equal to-- well, the negative squared is just going to be a positive 1. And you're going to get y squared is equal to x minus 3. So this is a situation here where for a given x, you could actually have 2 y-values. Let me show you. Let me attempt to sketch this graph. So let's say this is our y-axis. I guess I could call it this relation. This is our x-axis. And this right over here, y is a positive square root of x minus 3. That's going to look like this. So if this is x is equal to 3, it's going to look like this. That's y is equal to the positive square root of x minus 3. And this over here, y is equal to the negative square root of x minus 3, is going to look something like this. I should make it a little bit more symmetric looking, because it's going to essentially be the mirror image if you flip over the x-axis. So it's going to look something like this-- y is equal to the negative square root of x minus 3. And this right over here, this relationship cannot be-- this right over here is not a function of x. In order to be a function of x, for a given x it has to map to exactly one value for the function. But here you see it's mapping to two values of the function. So, for example, let's say we take x is equal to 4. So x equals 4 could get us to y is equal to 1. 4 minus 3 is 1. Take the positive square root, it could be 1. Or you could have x equals 4, and y is equal to negative 1. So you can't have this situation. If you were making a table x and y as a function of x, you can't have x is equal to 4. And at one point it equals 1. And then in another interpretation of it, when x is equal to 4, you get to negative 1. You can't have one input mapping to two outputs and still be a function. So in this case, the relation cannot-- for this relation, y cannot be represented as a mathematical function of x.