What is the range of the composite function h?

[Quadrat]

Homework Statement

[/B]
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?

Homework Equations

[/B]
##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##?

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##?

 

[Ray Vickson]

Homework Statement

[/B]
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?

Homework Equations

[/B]
##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##?

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##?

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.

 

[Quadrat]

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.

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?

 

[Ray Vickson]

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?

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.

 

[Mark44]

Throw away your calculator

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.