Question

How do I find the range of
y = sin(3x)^4 + 2

Answer 1

Is this \(\sin((3x)^4)\) or \((\sin(3x))^4\)?

Answer 2

(sin(3x))^4

Answer 3

Either way, you can find the range by using the fact that \(-1\le \sin(z)\le 1\).

Answer 4

did you try graphing it ?

Answer 5

I'm not going to take the easy way out by graphing

Answer 6

So the range of sin(x) i know is [-1,1], then you take those things to the 4th power to get 2 1's. What now?

Answer 7

Okay. First we have \(-1\le \sin(3x) \le 1\). Now take the fourth power of both sides:
\(0\le (\sin(3x))^4 \le 1 \implies 2\le (\sin(3x))^4+2\le 3.\)

Answer 8

So the range of \(y\) is \([2,3]\).

Answer 9

where did you get the 0?

Answer 10

When you take an even power of an inequality you have to be careful with numbers of different sign. Consider for example \(-5<1 \) does not imply \(25<1\). So you first have to divide the inequality into two parts, from -1 to 0 and then from 0 to 1. Do it this way and you'll get what I did.

Answer 11

okay, thanks

Answer 12

You're welcome!

Answer 13

did you study derivatives ?

Answer 14

No, I'm in algebra 2

Answer 15

oh...ok