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Worked example: domain and range from graph

Finding the domain and the range of a function that is given graphically. Created by Sal Khan.

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  • piceratops ultimate style avatar for user Anish Mantri
    What would I write if the function has arrows at the end of the line on both sides?
    (21 votes)
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  • mr pants teal style avatar for user Annei Titania
    How do you find the domain of a parabola? Do you use the same process?
    (12 votes)
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  • starky sapling style avatar for user Kayley
    I'm confused on what signs to use (greater than equal to, less than equal to, etc) I know that you use the greater than equal to and less than equal to, when it's included, but how do you know what sign to use when graphing? How do you know which way the graph is going? I'm not sure if I am making sense
    (6 votes)
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    • mr pink green style avatar for user David Severin
      The "equal" part of the inequalities matches the line or curve of the function, so it would be solid just as if the inequality were not there. Without the "equal" part of the inequality, the line or curve does not count, so we draw it as a dashed line rather than a solid line
      The < or > has to do with the shading of the graph, if it is >, shading is above the line, and < shading is below. The exception is a vertical line (x = #) where there is no above and below, so it changes to the left (<) or to the right (>)..
      So lets say you have an equation y > 2x + 3 and you have graphed it and shaded. If you try points such as (0,0) and substitute in for x and y, you get 0 > 3 which is a false statement, and if you did it right, shading would not go through this point. If point is (1,5) you can do the same thing, 5 > 5, but this would be right on the line, so the line would have to be dashed because this statement is not true either. One more point (0,6) would give 6>3 which is a true statement, and shading should include this point.
      Does this answer your question?
      (5 votes)
  • old spice man blue style avatar for user Lisa Barua
    What is a function?

    I keep confusing myself on what it is...
    I know domain is x and range is y
    (3 votes)
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  • blobby green style avatar for user 000ghostlyvenom
    -2<x<5 how can i write the inequalities?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      You would write your inequality in interval notation as:
      (-2, 5)
      The parentheses tell you that the inequalities do not include the end values of -2 and 5.

      If the inequality is: -2≤x≤5, then the interval notation is:
      [-2, 5]
      The square brackets tells you that the end values are included in the interval.

      If you have an inequality like: -2≤x<5, then the interval notation is:
      [-2, 5)
      A square bracket is on the -2 because it is included in the interval. The 5 gets a parentheses because it is not in the interval.

      Hope this helps.
      (6 votes)
  • aqualine ultimate style avatar for user Jairo
    range is bottom to top and domain is left to right.
    (3 votes)
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  • duskpin seedling style avatar for user Davin E
    what do I do if there are 2 points on one side of the domain and not a closed or open circle on the other side?
    (3 votes)
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  • duskpin ultimate style avatar for user sparkydoggo11
    How do you graph this domain? 0 is less than or equal to x, which is less than or equal to 20?
    PLEASE HELP ME!!
    (3 votes)
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  • blobby green style avatar for user MDESANTIAGO122
    how do you find the domain variable
    (2 votes)
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    • primosaur ultimate style avatar for user Buttcheeck Stinkleberg
      The domain of a function is the complete set of possible values of the independent variable. In plain English, this definition means: The domain is the set of all possible x-values which will make the function "work", and will output real y-values.
      When finding the domain, remember:
      The denominator (bottom) of a fraction cannot be zero
      The number under a square root sign must be positive in this section
      (2 votes)
  • piceratops ultimate style avatar for user Brandon R-S
    How would I write the range and the domain of the function y=1/x in interval notation? It's weird because x cannot equal 0, otherwise, the function would be undefined.
    (2 votes)
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Video transcript

The function f of x is graphed. What is its domain? So the way it's graphed right over here, we could assume that this is the entire function definition for f of x. So for example, if we say, well, what does f of x equal when x is equal to negative 9? Well, we go up here. We don't see it's graphed here. It's not defined for x equals negative 9 or x equals negative 8 and 1/2 or x equals negative 8. It's not defined for any of these values. It only starts getting defined at x equals negative 6. At x equals negative 6, f of x is equal to 5. And then it keeps getting defined. f of x is defined for x all the way from x equals negative 6 all the way to x equals 7. When x equals 7, f of x is equal to 5. You can take any x value between negative 6, including negative 6, and positive 7, including positive 7, and you just have to see-- you just have to move up above that number, wherever you are, to find out what the value of the function is at that point. So the domain of this function definition? Well, f of x is defined for any x that is greater than or equal to negative 6. Or we could say negative 6 is less than or equal to x, which is less than or equal to 7. If x satisfies this condition right over here, the function is defined. So that's its domain. So let's check our answer. Let's do a few more of these. The function f of x is graphed. What is its domain? Well, exact similar argument. This function is not defined for x is negative 9, negative 8, all the way down or all the way up I should say to negative 1. At negative 1, it starts getting defined. f of negative 1 is negative 5. So it's defined for negative 1 is less than or equal to x. And it's defined all the way up to x equals 7, including x equals 7. So this right over here, negative 1 is less than or equal to x is less than or equal to 7, the function is defined for any x that satisfies this double inequality right over here. Let's do a few more. The function f of x is graphed. What is its range? So now, we're not thinking about the x's for which this function is defined. We're thinking about the set of y values. Where do all of the y values fall into? Well, let's see. The lowest possible y value or the lowest possible value of f of x that we get here looks like it's 0. The function never goes below 0. So f of x-- so 0 is less than or equal to f of x. It does equal 0 right over here. f of negative 4 is 0. And then the highest y value or the highest value that f of x obtains in this function definition is 8. f of 7 is 8. It never gets above 8, but it does equal 8 right over here when x is equal to 7. So 0 is less than f of x, which is less than or equal to 8. So that's its range. Let's do a few more. This is kind of fun. The function f of x is graphed. What is its domain? So once again, this function is defined for negative 2. Negative 2 is less than or equal to x, which is less than or equal to 5. If you give me an x anywhere in between negative 2 and 5, I can look at this graph to see where the function is defined. f of negative 2 is negative 4. f of negative 1 is negative 3. So on and so forth, and I can even pick the values in between these integers. So negative 2 is less than or equal to x, which is less than or equal to 5.