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Range of composite function

  1. Nov 21, 2016 #1
    1. The problem statement, all variables and given/known data

    The function ##f##, ##{f: ℤ → ℚ}## defined as ##f(a)=cos(πa)##
    The function ##g##, ##{g: ℚ→ ℝ}## defined as ##g(a)=(5a)/4##

    Let h be the composite funciton ##h(a)=f(g(a))##

    What's the range of this function h?

    2. Relevant equations

    ##h(a)=cos(5πa/4)##

    The domain of ##h## should be ##ℤ## and ##ℝ## its codomain. ##{h: ℤ → ℝ}##.

    So a must be an integer, right? How do I sort out the range of ##h##?

    3. The attempt at a solution

    This is just the last step in a homework assignment
    So ##a## must be an integer, right? So any number ##n∈ℤ## in ##h## can be used. I tried with integers up to 10 to see what values I'd get. I just don't know how to go on with this one. How do I sort out the range of ##h##?
     
  2. jcsd
  3. Nov 21, 2016 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Try to get a feeling for what is going on by testing a few small values such as ##a = 0, 1, 2, 3## to see what you get.
     
  4. Nov 21, 2016 #3
    That's what I did. Starting from ##a=0## to ##a=15##
    ##h(0)=1##
    ##h(1)=-1/sqrt(2)##
    ##h(2)=0##
    ##h(3)=1/sqrt(2)##
    ##h(4)=-1##
    ##h(5)=1/sqrt(2)##
    ##h(6)=5*E(-13)##
    ##h(7)=-1/sqrt(2)##
    ##h(8)=1##
    ##h(9)=-1/sqrt(2)##
    ##h(10)=-5*E(-13)##
    ##h(11)=1/sqrt(2)##
    ##h(12)=-1##
    ##h(13)=1/sqrt(2)##
    ##h(14)=1,5*E(-12)##
    ##h(15)=1/sqrt(2)##

    Still I can't figure out what the range is. Especially when I get values like h(14), h(26), h(30) etc. What am I missing?
     
  5. Nov 21, 2016 #4

    Ray Vickson

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    Science Advisor
    Homework Helper

    Throw away your calculator; you don't need it in this problem, and its use is just confusing you. Things like ##5 E(-13)## are rounded versions of ##0## exactly. You should know---without ever consulting a calculator---what are cosines of angles like 0, ##\pi##, ##2 \pi##, ##3\pi##, etc., as well as for angles like ##\pi/4##, ##2\pi/4 = \pi/2##, ##3 \pi/4##, etc.
     
  6. Nov 21, 2016 #5

    Mark44

    Staff: Mentor

    Yes, absolutely. In addition to the angles Ray listed, you should know, by heart, the trig functions of ##\pi/6, \pi/3, 2\pi/3, 5\pi/6## and their corresponding angles in the 3rd and 4th quadrants.
     
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