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Worked example: positive & negative intervals

Finding the positive or negative intervals of a function from its graph. Created by Sal Khan.

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  • primosaur ultimate style avatar for user Shimmy
    What I understand is that the result returned by f(x) completely ignores the y value. Which is obviously wrong, otherwise the right answer to this question is if the plot is anywhere to the left of X=0.
    So can you please clarify that to me?
    (14 votes)
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    • blobby green style avatar for user Mark Crawford
      The function of x {f(x)} at any given point would be a coordinate plotted on the y-axis. You should consider them interchangeable. Any output derived from an input plotted along the horizontal x-axis can be, and usually is, plotted along the vertical y-axis. If you want to learn how to do these problem sets, just ask yourself if it's below the x-axis or not. If it's below, then it's less than zero. If it is above, then it is greater than zero.
      (6 votes)
  • aqualine ultimate style avatar for user edwards46866
    I don't understand what "F" is? Is it a variable or is something I haven't learn't?
    (5 votes)
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  • duskpin ultimate style avatar for user michelle wang 🍑
    I don't understand what the "graph is below the x-axis" means.
    (5 votes)
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  • starky seedling style avatar for user Emlyn Vo
    This doesn't cover how to do the equations, such as -1 > x > -5.
    (5 votes)
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  • purple pi purple style avatar for user louisaandgreta
    In the following exercise “Positive and negative intervals” I have an issue understanding the terminology used. Sometimes they refer in the question to the function as f(x) and in the graph tells you that it’s equal to y, pretty traditional, but sometimes they refer in the question to the function as f.
    Is there a difference?
    Especially when it comes to which axis you need to look at to see the intervals over which the function is positive or negative.

    For instance they’ll say “select the interval where h is positive” but other times they’ll say “select the interval where g(x)<0”
    (2 votes)
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    • mr pink green style avatar for user David Severin
      So they are strongly related, but have different purposes. If you are talking about the function, you can use just the letter to distinguish it from other functions that may be in the same problem or you could say f(x). However, if you write it in the equation, you have to use f(x) which is read as f of x and means the function f in terms of x. So one is talking about the function and the other is expressing the function in an equation. Thus f, f(x) and y are all the same and are used for different purposes. So both of your quoted requests are the same and asking where domain (x) meets some criteria.
      (3 votes)
  • piceratops ultimate style avatar for user Prathamesh Sawant
    How to write x>2 in interval notation?
    (2 votes)
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  • blobby green style avatar for user sulinskyp
    What are the real life examples of one to one function?
    (3 votes)
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    • blobby green style avatar for user Mark Crawford
      You mean a slope of one? Like the y=x? Think of any relationship where one value is always equal to another. If I can buy one candy bar from the girl scouts for one dollar, and I then plotted the function of number of candy bars bought and the price of that amount of candy bars, it would be a y=x. Being such a simple function, it seems trivial to graph it when one value is equal to another, but it's good practice for slope, and can help prepare for more difficult functions. What if I could get five pieces of gum for a dollar? What would that function be?

      y=x5 (y is the dollar amount spent. For each dollar, you have five times as many pieces of gum.)
      (0 votes)
  • blobby green style avatar for user ch04195
    I dont understand what this means 2<x<3. How do you know where that is on the graph?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      2<x<3 can be broken into 2 parts:
      2<x: 2 is less than x
      x<3: x is less than 3
      When you put these together you get:
      2 is less than x and x is less than 3
      This means x is an fractional or decimal value located between 2 and 3.

      When looking at the graph, look at the x-axis for the value between 2 and 3. Then look at what is happening on the graph for than set of numbers.

      Hope this helps.
      (2 votes)
  • female robot grace style avatar for user Kornelija
    What if the quadratic function never touches the x-axis? When is a graph positive for all real values of x then?
    (1 vote)
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  • aqualine ultimate style avatar for user Aurora Borealis
    In this video, all of the examples say "f(x)," which obviously refers to the x axis. However, in the current practice problems for this skill(and many other problems related to functions), the problems have functions such as y=g(x). Can someone explain what this means?
    (1 vote)
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    • mr pink green style avatar for user David Severin
      f(x), g(x), h(x), etc. is functional notation and is read as f of x. This is related to y, so it refers to the value in the y direction, not the x direction. It answers the question what is the value of a function at a given x (f(3), f(-5), etc.)? You generally do not worry about y = g(x) because you are substituting the functional notation in for y. You learn it now, but the best applications will come in Algebra II in combining functions.
      (1 vote)

Video transcript

A function, f of x, is plotted below. Highlight an interval where f of x is less than 0. So f of x-- which is really being plotted on the vertical axis right over here-- x is the horizontal axis. f of x being less than 0 really means that the graph is below the x-axis. So the function is negative in this interval right over here and this interval over here. So I could put this anywhere right over here, or I could stick it anywhere right over here. Let me stick it right over here. There we go. Got it right. Let's do a couple more. So the function is plotted below. Highlight an interval where f of x is greater than 0. So I could do this area right over here where the function is above the x-axis, or I could do this area right over here where the function goes way above the x-axis. Well, it even goes off the page. So let's stick it right over here. Let's do one more. So highlight an interval where f of x is greater than 0. Once again, I can do this region right over here where the function is above the x-axis, or over here where it's above the x-axis. I'll do it here just for fun. There we go.