# Range of composite function

1. Nov 21, 2016

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. Nov 21, 2016

### Ray Vickson

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.

3. Nov 21, 2016

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?

4. Nov 21, 2016

### Ray Vickson

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.

5. Nov 21, 2016

### 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.