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Does a vertical line represent a function?

Explaining why a vertical line doesn't represent a function. Created by Sal Khan.

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  • duskpin ultimate style avatar for user Ronit Patel
    can a horizontal line represent a function?
    (16 votes)
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  • piceratops ultimate style avatar for user Teddy Karrer
    could he have just used the vertical line test?
    (10 votes)
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    • leaf red style avatar for user Noble Mushtak
      Yes, he could've. If he did that, then he would've noticed that the relation intersects the vertical line x=-2 at infinitely many points. This is because the relation is x=-2, so obviously it intersects it at infinitely many points.

      However, I think Sal was trying to demonstrate a more rigorous way of testing a relation for being a function. Instead of just doing a vague, vertical line test, he used the definition of a function to test the relation for being a function.

      I hope this helps!
      (37 votes)
  • leaf green style avatar for user areesha.kadiwal
    Can there be many domain but getting only one range? If the line will be horizontal will it be a function?
    (4 votes)
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    • leaf blue style avatar for user Stefen
      One domain and one range although the domain can consist of the union of various regions on the x-axis. EG {-100 < x < -10} U {-1 < x < 1} U {10 < x < 100}. the U is the symbol of union.
      A horizontal line is a function, but a pretty boring one since no matter what x value you input, the output will always be the same. EG f(x)=5. No matter what x is, the output is always 5. As you can see, the output value does not depend on the input value x.
      (2 votes)
  • purple pi pink style avatar for user Amy D
    So is Sal saying that x -> f(x) -> infinity is not a function? If you just wrote the infinity sign could it be considered as only one output?
    (2 votes)
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  • aqualine ultimate style avatar for user Haya Aboudan
    is a function with multiple outputs a logarithm?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      No. A function, by definition, can not have multiple outs for a specific input value. Each input can create only one output to be a function. Thus, any equation that doesn't meet this definition would not be a function.

      FYI.. there are logarithmic functions.
      (4 votes)
  • blobby green style avatar for user ursaciucantonio
    3.14159, he actually started defining pi :)
    (3 votes)
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  • blobby blue style avatar for user Johnny
    Would a drastically curved line on a graph represent a function? Does the curve have to go through x and y to be a function?
    (1 vote)
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    • purple pi purple style avatar for user RJacob1
      For a relation to be a function, use the Vertical Line Test: Draw a vertical line anywhere on the graph, and if it never hits the graph more than once, it is a function. If your vertical line hits twice or more, it's not a function. For example, a circle is not a function because when you draw a vertical line on top of its graph, the vertical line will cut through the circle twice.
      (5 votes)
  • hopper jumping style avatar for user Sam D
    What's the vertical and horizontal line test?
    (1 vote)
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    • mr pink green style avatar for user David Severin
      The vertical line test is used to determine if a graph of a relationship is a function or not. if you can draw any vertical line that intersects more than one point on the relationship, then it is not a function. This is based on the fact that a vertical line is a constant value of x, so if there is one input, x, with more than two outputs, y, then it breaks the function rule. A horizontal line test does not have as much meaning.
      (4 votes)
  • male robot hal style avatar for user Leo
    Can a vertical line be represented by an equation?
    (2 votes)
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  • leafers seed style avatar for user Gracie Lawyer
    does anyone know how to solve this? :
    "solve by graphing and check" y= -2x-2 and y=-4 ??
    (2 votes)
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    • hopper cool style avatar for user Seed Something
      Graph to solve: y = -2x -2 and y = -4

      y = -2x -2

      Is in Slope Intercept Form, which means we can look at the equation and see the slope of the line is -2, (because it's the coefficient of x), the second term is the y intercept and also -2, (where the line crosses the y-axis).

      y-intercept of -2, means…
      when x is zero, y is -2.
      So we're able to graph that Point.
      Point at: (0, -2) ←y intercept

      ★ Since the Slope is Negative, -2, we know the line is decreasing, meaning it goes ↘️ downward from left to right, and…

      •A Slope of -2, means every time we move one unit Right, we move two Down.

      ★So let's do that from the y intercept!
      If we're at (0, -2) and need to…

      move one to the Right,
      that's plus one on x axis…
      x = 0 + 1 = 1

      and for y we need two Down,
      that's minus 2 on y axis…
      y = -2 -2 = -4

      Now we have another coordinate on the same line!
      Point at: (1, -4)

      ★(0, -2) and (1, -4)
      Draw a line through both coordinate Points, to finish graphing the first equation.

      • Now to graph…
      y = -4
      Notice there is no written slope in this equation, just a y intercept, this is because it's a…
      ↔️ a Horizontal Line, slope = 0
      It runs flat crossing y-axis at -4.
      Find -4 on y-axis, draw a straight, side to side, ↔️ Horizontal Line through it.

      Look to see where these two lines cross.
      That (x, y) coordinate is the Point of Intersection between these two lines, the only x and y coordinate pair that make both of these equations True Statements.

      Continued in Comments
      (2 votes)

Video transcript

In the following graph, is y a function of x? So in order for y to be a function of x, for any x that you input into the function, any x for which the function is defined. So let's say we have y is equal to f of x. So we have our little function machine. It should spit out exactly one value of y. If it spits out multiple values of y, we don't know what f of x is going to be equal to. It could be equal to any of those possible values for y. So let's see if, for this graph, whether for a given x it spits out exactly one y. Well, the function seems to be only defined so the domain of this function is x is equal to negative 2. That's the only place where we have a definition for it. And if we try to input negative 2 into this little black box, what do we get? Do we get exactly one thing? No. If we put in negative 2 here, we could get anything. The point negative 2, 9 is on this relation. Negative 2, 8 is on this relation. Negative 2, 7; negative 2, 7.5; negative 2, 3.14159-- they're all on these. So if you put a negative 2 into this relation, essentially, you actually get an infinite set of values. It could be 9. It could be 3.14. It could be 8. It could be negative 8. You get an infinite number of results. So since it does not map to exactly one output of this function, in the following graph, y is not a function of x.