Introduction to piecewise functions | Algebra (video) | Khan Academy (2024)

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  • Marko Arezina

    9 years agoPosted 9 years ago. Direct link to Marko Arezina's post “Where in mathematics woul...”

    Where in mathematics would you see piecewise functions?

    (62 votes)

    • Stefen

      9 years agoPosted 9 years ago. Direct link to Stefen's post “Where ever input threshol...”

      Introduction to piecewise functions | Algebra (video) | Khan Academy (4)

      Introduction to piecewise functions | Algebra (video) | Khan Academy (5)

      Introduction to piecewise functions | Algebra (video) | Khan Academy (6)

      Where ever input thresholds (or boundaries) require significant changes in output modeling, you will find piece-wise functions.
      In your day to day life, a piece wise function might be found at the local car wash: $5 for a compact, $7.50 for a midsize sedan, $10 for an SUV, $20 for a Hummer.
      Or perhaps your local video store: rent a game, $5/per game, rent 2-3 games, $4/game, rent more than 5 games, $3/per game.
      Ask your folks about tax brackets, another piece-wise function.
      Hmmm, something more scientific? How about modeling the fuel usage of a space shuttle from launch to docking with the ISS. Each phase, launch, staging, orbit insertion, course correction and docking is a piece that has a very different characteristics of fuel consumption, and will require a different expression with different variables (air resistance, weight, gravity, burn rates etc.) at each stage in order to model it correctly.
      So this piece wise stuff may seem arcane or just a very special (infrequent) case, but it is not, it is a fixture in the mathematical landscape, so enjoy the view!
      Keep Studying!

      (233 votes)

  • Josh (SpongeJr)

    9 years agoPosted 9 years ago. Direct link to Josh (SpongeJr)'s post “Edit: The Algebra I secti...”

    Edit: The Algebra I section has been expanded to include some modules that fill in these gaps nicely, and a few others.

    Great job, Khan Academy! I am enjoying the new exercises, and I feel they really help fill in some small gaps that were there in the content. I believe those new modules added significant value to the lessons in that section of the KA content.

    These are a couple of the new modules added to address this:
    https://www.khanacademy.org/math/algebra/algebra-functions/domain-and-range/e/domain-of-algebraic-functions
    https://www.khanacademy.org/math/algebra/algebra-functions/piecewise_functions/e/evaluating-piecewise-functions

    There were about 10 new modules added in the Algebra I "Functions" section, I believe. Very nice!

    Kudos, and thanks!
    (old question kept for historical purposes)
    -------------------------
    There is an exercise in the Algebra I content-- "Domain of a function" ( https://www.khanacademy.org/mission/algebra/task/6652614144688128 ).

    One or more of the questions is all about the domain for a piecewise function. The Hint text says, "f(x) is a piecewise function, so we need to examine where each piece is undefined." (and goes on from there).

    I have been looking and looking for Algebra I content that mentions piecewise functions, to make sure I learn it at the earliest point that I should have learned it. I have only been able to find it in the Algebra II lessons. It's interesting (and kind of cool) that this video just came out as I've been looking for it.

    Is this the first time piecewise functions are explained in the Khan Academy lessons? If so, I think some of the problems in the set I linked, or at least the Hint text for them, might be out of place. If not, can anyone point me to a lesson where they are explained or at least mentioned earlier than the "Domain of a function" lesson in Algebra I? I can't find them mentioned on this playlist, for example: https://www.khanacademy.org/math/algebra/algebra-functions -- and definitely not anywhere earlier than the exercise I mentioned.

    Thanks!

    (38 votes)

  • anjalinc3

    8 years agoPosted 8 years ago. Direct link to anjalinc3's post “Does the order in which y...”

    Does the order in which you list the different pieces of the function matter? If so, would you go from least to greatest x-values or y-values?

    (12 votes)

    • Dominik.Ehlert

      8 years agoPosted 8 years ago. Direct link to Dominik.Ehlert's post “No, you can order the pie...”

      Introduction to piecewise functions | Algebra (video) | Khan Academy (14)

      No, you can order the pieces as you like. But usually you will find the order from the least to the greatest x-values, so you can use it as instructions from the left of the right in die graph.

      (16 votes)

  • Cowboy

    6 years agoPosted 6 years ago. Direct link to Cowboy's post “Why did Sal put the y coo...”

    Why did Sal put the y coordinates before the x coordinates in his function? Is this going to give a wrong coordinate in the final output?
    EX— -9, -9 < x ≤ -5

    (13 votes)

    • Ninja

      6 years agoPosted 6 years ago. Direct link to Ninja's post “You could have done it in...”

      You could have done it in any order as only the end product counts because if you read it left to right it will say, draw a line at y=-9 with a domain of -9 to -5 only including -5.

      (7 votes)

  • Stopmotiongeek

    8 years agoPosted 8 years ago. Direct link to Stopmotiongeek's post “Wait! At 1:35 Sal defines...”

    Wait! At

    Introduction to piecewise functions | Algebra (video) | Khan Academy (19) 1:35

    Sal defines the first function as (-9, -9<x≤-5]. Now from my understanding the input or domain is still the x variable and the output or y. Although this is arranged with the y variable first and then the x variable. Why not x,y? (-9<x≤-5,-9)?

    (8 votes)

    • Nikolay

      4 years agoPosted 4 years ago. Direct link to Nikolay's post “What he's saying is that ...”

      What he's saying is that the output is -9 when -9<x≤-5. Perhaps the inclusion of the word could have avoided confusion.

      (3 votes)

  • Shreya

    9 years agoPosted 9 years ago. Direct link to Shreya's post “I tried solving the exerc...”

    I tried solving the exercise for piecewise functions. But the hints to the answers talk about the point being hollow and filled. I mean there is an exercise even before a video to explain the content. So is there is a video regarding this under another section?

    (5 votes)

    • Karah Han

      9 years agoPosted 9 years ago. Direct link to Karah Han's post “This video shows a bit ho...”

      This video shows a bit how to use open and closed circles. https://www.khanacademy.org/math/algebra-basics/core-algebra-graphing-lines-slope/core-algebra-graphing-linear-inequalities/v/solving-and-graphing-linear-inequalities-in-two-variables-1
      An open circle means "Does not include this value" (so like < & >). A closed circle means "Also includes this point" (like <= & >=). A good way to remember is that an open circle (○) is not colored in so the point it is on is NOT included. A closed circle (•) is colored in so it INCLUDES the point it is on. I hope this answers your question. Let me know if it doesn't.

      (8 votes)

  • Rishi

    9 years agoPosted 9 years ago. Direct link to Rishi's post “Can't you just do the ver...”

    Can't you just do the vertical line test on two of those little parts, and prove that this is not a function?

    (5 votes)

    • nithanth9

      9 years agoPosted 9 years ago. Direct link to nithanth9's post “We can't use the vertical...”

      We can't use the vertical line test because there is more than one line. To use the vertical line test, the relation needs to be continuous(all the dots on a line are connected by one line). Since piecewise-functions are discontinuous, you can not use the vertical line test.
      Hope this helps. :)

      (7 votes)

  • Abdullah Sarfraz

    a year agoPosted a year ago. Direct link to Abdullah Sarfraz's post “why does sal put -9, 6 an...”

    why does sal put -9, 6 and -7 at the left of the equation. What does it represent and mean??

    (7 votes)

  • jones166

    6 years agoPosted 6 years ago. Direct link to jones166's post “what confuses me is the w...”

    what confuses me is the whole thing anyone care to slow it down for me thank you

    (6 votes)

    • Pogo

      4 years agoPosted 4 years ago. Direct link to Pogo's post “Kinda late, but you can a...”

      Kinda late, but you can adjust the speed in the video settings.

      (2 votes)

  • Josiah

    6 years agoPosted 6 years ago. Direct link to Josiah's post “can the pieces ever be ve...”

    can the pieces ever be vertical?

    (1 vote)

    • Kim Seidel

      6 years agoPosted 6 years ago. Direct link to Kim Seidel's post “No... vertical lines are ...”

      Introduction to piecewise functions | Algebra (video) | Khan Academy (36)

      No... vertical lines are not functions. Since you are working with piecewise functions, all the pieces need to be functions.

      (10 votes)

Video transcript

- [Voiceover] By now we're used to seeing functions defined like h(y)=y^2 or f(x)= to the square root of x. But what we're now going to explore is functions that aredefined piece by piece over different intervalsand functions like this you'll sometimesview them as a piecewise, or these types of function definitions they might be called apiecewise function definition. Let's take a look at thisgraph right over here. This graph, you can see that the function is constant over this interval, 4x. And then it jumps upin this interval for x, and then it jumps back downfor this interval for x. Let's think about how we would write this using our function notation. If we say that this rightover here is the x-axis and this is the y=f(x) axis. Then, let's see, our functionf(x) is going to be equal to, there's three different intervals. So let me give myself some space for the three different intervals. Now this first intervalis from, not including -9, and I have this open circle here. Not a closed in circle. So not including -9 butx being greater than -9 and all the way up to and including -5. I could write that as -9 is less than x, less than or equal to -5. That's this interval, and what is the value of the functionover this interval? Well we see, the valueof the function is -9. It's a constant -9 over that interval. It's a little confusing because the value of the function is actually also the value of the lower bound on thisinterval right over here. It's very important to look atthis says, -9 is less than x, not less than or equal. If it was less than orequal, then the function would have been defined atx equals -9, but it's not. We have an open circle right over there. But now let's look at the next interval. The next interval isfrom -5 is less than x, which is less than or equal to -1. Over that interval, thefunction is equal to, the function is a constant 6. It jumps up here. Sometimes people call this astep function, it steps up. It looks like stairs to some degree. Now it's very importanthere, that at x equals -5, for it to be defined only one place. Here it's defined by this part. It's only defined over here. So that's why it'simportant that this isn't a -5 is less than or equal to. Because then if you put-5 into the function, this thing would be filled in, and then the function wouldbe defined both places and that's not cool for a function, it wouldn't be a function anymore. So it's very important that when you input - 5 in here, you know whichof these intervals you are in. You can't be in two of these intervals. If you are in two of these intervals, the intervals shouldgive you the same values so that the function maps, from one input to the same output. Now let's keep going. We have this lastinterval where we're going from -1 to 9. >From -1 to +9. And x starts off with -1 less than x, because you have an opencircle right over here and that's good because X equals -1 is defined up here, all the way to x isless than or equal to 9. Over that interval, what isthe value of our function? Well you see, the value ofour function is a constant -7. A constant -7 and we're done. We have just constructed a piece by piece definitionof this function. Actually, when you see thistype of function notation, it becomes a lot clearer why function notation is useful even. Hopefully you enjoyed that. I always find these piecewisefunctions a lot of fun.

Introduction to piecewise functions | Algebra (video) | Khan Academy (2024)
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