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Math Help - Piecewise Function - Domain & Range

  1. #1
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    Piecewise Function - Domain & Range

    For the function
    _____{x+3 if -5≤x<2
    f(x)={x^2 if 2≤x≤4

    I have to sketch the graph, determine the domain and range of f(x), and determine f(2). Help please?
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  2. #2
    MHF Contributor red_dog's Avatar
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    f(x)=\left\{\begin{array}{ll}x+3 & ,-5\leq x<2\\x^2 & ,2\leq x\leq 4\end{array}\right.

    The domain is [-5,4].

    To find the range we can use the following statement abou function:

    If f:A\to B and X, \ Y are two subsets of A, then

    f(X\cup Y)=f(X)\cup f(Y)

    In this case let X=[-5,2), \ Y=[2,4]

    We have f(X)=[-2,5), \ f(Y)=[4,16].

    Then the range is [-2,5)\cup[4,16]=[-2,16]
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  3. #3
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    Quote Originally Posted by red_dog View Post
    f(x)=\left\{\begin{array}{ll}x+3 & ,-5\leq x<2\\x^2 & ,2\leq x\leq 4\end{array}\right.

    The domain is [-5,4].

    To find the range we can use the following statement abou function:

    If f:A\to B and X, \ Y are two subsets of A, then

    f(X\cup Y)=f(X)\cup f(Y)

    In this case let X=[-5,2), \ Y=[2,4]

    We have f(X)=[-2,5), \ f(Y)=[4,16].

    Then the range is [-2,5)\cup[4,16]=[-2,16]
    Wow... That is completely confusing. Never seen anything that looks anything like that stuff.
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    MHF Contributor red_dog's Avatar
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    Quote Originally Posted by BeSweeet View Post
    Wow... That is completely confusing. Never seen anything that looks anything like that stuff.
    Then let's do it in another way.

    -5\leq x<2

    Add 3 to all members: -2\leq x+3<5

    Then, if x\in[-5,2) then f(x)\in[-2,5)

    2\leq x\leq 4

    Square all members: 4\leq x^2\leq 16

    Then, if x\in[2,4] then f(x)\in[4,16].

    So, the range is [-2,5)\cup[4,16]=[-2,16]

    Is it better now?
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  5. #5
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    Quote Originally Posted by BeSweeet View Post
    For the function
    _____{x+3 if -5≤x<2
    f(x)={x^2 if 2≤x≤4

    I have to sketch the graph...
    Sketch y = x + 3, and then erase everything before x = -5 and after x = 2. Make sure to draw a filled-in circle for the left-hand endpoint and an "open" circle for the right-hand endpoint.

    Then sketch y = x^2, and erase everything before x = 2 and after x = 4. Make sure to draw filled-in circles for each of the endpoints.

    Quote Originally Posted by BeSweeet View Post
    ...determine the domain and range of f(x)...
    The domain is given: it's the x-values for which the function is defined.

    To find the range, look at your graph. Which y-values are covered by this graph? (If you "collapsed the graph sideways onto the y-axis, which portions would be covered?)

    Quote Originally Posted by BeSweeet View Post
    ...and determine f(2).
    The function is defined for x = 2. So look at the function rule, find the half which is defined for x = 2, and plug 2 in for x in that half's rule.

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    Quote Originally Posted by stapel View Post
    Sketch y = x + 3, and then erase everything before x = -5 and after x = 2. Make sure to draw a filled-in circle for the left-hand endpoint and an "open" circle for the right-hand endpoint.

    Then sketch y = x^2, and erase everything before x = 2 and after x = 4. Make sure to draw filled-in circles for each of the endpoints.


    The domain is given: it's the x-values for which the function is defined.

    To find the range, look at your graph. Which y-values are covered by this graph? (If you "collapsed the graph sideways onto the y-axis, which portions would be covered?)


    The function is defined for x = 2. So look at the function rule, find the half which is defined for x = 2, and plug 2 in for x in that half's rule.

    I'm still unsure at how you are supposed to graph this thing. I don't understand the domain & range either. I do understand the f(2) thing. The answer for that part is 4, right?
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  7. #7
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    Quote Originally Posted by BeSweeet View Post
    I'm still unsure at how you are supposed to graph this thing.
    To learn how to graph linear equations, such as y = x + 3, try here.

    to learn how to graph quadratics, such as y = x^2, try here.

    They were supposed to have covered this material way before moving on to piecewise functions, etc.
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  8. #8
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    Quote Originally Posted by stapel View Post
    To learn how to graph linear equations, such as y = x + 3, try here.

    to learn how to graph quadratics, such as y = x^2, try here.

    They were supposed to have covered this material way before moving on to piecewise functions, etc.
    I get it!
    a. Figured out how to sketch it.
    bA. For the domain, is it (-, -5)∪(-5, 2)∪(2, 4)∪(4, )... Something like that?
    bB. For the range, I'm still not getting that part.
    c. Determining f(2) is simple. Should be 4, right?
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