ATAR Notes: Forum

VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: Zebra on April 21, 2011, 09:08:46 pm

Title: Implied domain and range
Post by: Zebra on April 21, 2011, 09:08:46 pm: We are told that the domain of Cos(x) is 0 to pie

Cos(sin-1(x))

the implied domain?

Going through questions I got wrong in Exercise 3C is driving me mad!
Title: Re: Implied domain and range
Post by: xZero on April 21, 2011, 09:18:02 pm: since the range of sin^-1(x) is -pi/2 to pi/2, for the function Cos(sin^-1(x)) to exist, we must restrict the domain of sin^-1(x) to x=0 to x=1. Hence the implied domain is 0≤x≤1
Title: Re: Implied domain and range
Post by: Vincezor on April 22, 2011, 07:42:17 pm: So, we need find the implied domain of $cos[sin^{-1}(x)]$

as you said, $dom cos(x) = [0,\pi]$

You may remember from composite functions in methods that $Ran g \subset Dom f$

$Ran sin^{-1} = Dom sin$ = $[-\frac{\pi}{2},\frac{\pi}{2}]$

However we must restrict this range so that it becomes a subset of the domain of cos(x), so we get

0 ≤ $sin^{-1}(x)$ ≤ $\frac{\pi}{2}$

EDIT: Solve for x

sin(0) ≤ x ≤ sin( $\frac{\pi}{2}$ )

which is

0 ≤ x ≤ 1

Is this working out right? :S
Title: Re: Implied domain and range
Post by: luffy on April 22, 2011, 08:44:58 pm: Quote from: Vincezor on April 22, 2011, 07:42:17 pm
So, we need find the implied domain of $cos[sin^{-1}(x)]$

as you said, $dom cos(x) = [0,\pi]$

You may remember from composite functions in methods that $Ran g \subset Dom f$

$Ran sin^{-1} = Dom sin$ = $[-\frac{\pi}{2},\frac{\pi}{2}]$

However we must restrict this range so that it becomes a subset of the domain of cos(x), so we get

0 ≤ $sin^{-1}(x)$ ≤ $\frac{\pi}{2}$

Multiply everything by $sin(x)$ so middle becomes x

sin(0) ≤ x ≤ sin( $\frac{\pi}{2}$ )

which is

0 ≤ x ≤ 1

Is this working out right? :S

Yes - ur working is correct. Except, your not multiplying everything by sin....
Title: Re: Implied domain and range
Post by: Vincezor on April 22, 2011, 09:01:30 pm: Quote from: luffy on April 22, 2011, 08:44:58 pm

Yes - ur working is correct. Except, you're (:P) not multiplying everything by sin....

Oh my bad, was I meant to say I was solving for x? Because I think that's what I was meant to do...
Title: Re: Implied domain and range
Post by: luffy on April 22, 2011, 09:34:40 pm: Quote from: Vincezor on April 22, 2011, 09:01:30 pm
Quote from: luffy on April 22, 2011, 08:44:58 pm

Yes - ur working is correct. Except, you're (:P) not multiplying everything by sin....

Oh my bad, was I meant to say I was solving for x? Because I think that's what I was meant to do...

LOL! - Haha... I actually laughed at how you edited my quote. Good one!

P.S. I would love to know you in real life :P :P :P