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March 28, 2024, 11:08:50 pm

Author Topic: VCE Specialist 3/4 Question Thread!  (Read 2164195 times)  Share 

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ArtyDreams

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9615 on: March 09, 2020, 10:42:38 am »
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Thank you so much for this detailed explanation! I'm kind of getting it now :)

dream chaser

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9616 on: March 14, 2020, 01:26:40 pm »
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Hi Guys,

How do you find the domain and implied range for:

g(x) = x/(|x|-4)

I know the domain is that: (|x|-4) ≠0

Is the best way to do it is to draw y=|x|-4 and then find the domain?

All help will be much appreciated. Thanks  :)

S_R_K

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9617 on: March 14, 2020, 03:54:29 pm »
+2
Hi Guys,

How do you find the domain and implied range for:

g(x) = x/(|x|-4)

I know the domain is that: (|x|-4) ≠0

Is the best way to do it is to draw y=|x|-4 and then find the domain?

All help will be much appreciated. Thanks  :)

The maximal domain is all reals except for the solutions to |x|=4. No need to sketch for this.

To find the range, I'd case-break and sketch the graph:

Case 1: For x ≥ 0, x/(x – 4) = 1 + 4/(x – 4).

Case 2: For x < 0, x/(-x – 4) = -1 + 4/(x+4).

TheEagle

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9618 on: March 14, 2020, 03:55:05 pm »
+2
Hi Guys,

How do you find the domain and implied range for:

g(x) = x/(|x|-4)

I know the domain is that: (|x|-4) ≠0

Is the best way to do it is to draw y=|x|-4 and then find the domain?

All help will be much appreciated. Thanks  :)


I would do this:

IxI - 4 = 0
IxI = 4

if x<0, then:  -x=4    therefore     x=-4

If x>=0, then:  x=4

therefore, domain:   R\{-4,4}

dream chaser

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9619 on: March 14, 2020, 04:08:52 pm »
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Thanks TheEagle and S_R_K for the help. It is much appreciated.  :)

Also for f(x) = √ (x² -1), I need to find the maximal domain and range

I found the maximal domain to be x∈ (-∞, -1] U [1, ∞).

How would I find the range? I know we can do it with calculus. Is there any other way apart from calculus to find the range for f(x)? If so, how would you do it.

All help will be much appreciated. Thanks  :)
« Last Edit: March 14, 2020, 04:45:59 pm by dream chaser »

S_R_K

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9620 on: March 14, 2020, 04:59:13 pm »
+1
Thanks TheEagle and S_R_K for the help. It is much appreciated.  :)

Also for f(x) = √ (x² -1), I need to find the maximal domain and range

I found the maximal domain to be x∈ (-∞, -1] U [1, ∞).

How would I find the range? I know we can do it with calculus. Is there any other way apart from calculus to find the range for f(x)? If so, how would you do it.

All help will be much appreciated. Thanks  :)

The range of f(g(x)) = the range of f(u), where the domain of u is the intersection between the range of g(x) and the maximal domain of f(x).

So in this case, the intersection between the range of x^2 – 1 and the maximal domain of √x is [0, ∞). So the range of √ (x² -1) is the range of √u, where u is in [0, ∞).

dream chaser

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9621 on: March 14, 2020, 05:05:35 pm »
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The range of f(g(x)) = the range of f(u), where the domain of u is the intersection between the range of g(x) and the maximal domain of f(x).

So in this case, the intersection between the range of x^2 – 1 and the maximal domain of √x is [0, ∞). So the range of √ (x² -1) is the range of √u, where u is in [0, ∞).

Sorry S_R_K, I should have made it clear. The question regarding f(x) has got nothing to do with g(x). It is a totally separate question

S_R_K

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9622 on: March 14, 2020, 05:11:52 pm »
+1
Sorry S_R_K, I should have made it clear. The question regarding f(x) has got nothing to do with g(x). It is a totally separate question

Yes, my reply was written with that understanding. In my reply I was taking g(x) to be x^2 – 1 and f(x) = √x, and hence √(x^2 – 1) = f(g(x)).

dream chaser

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9623 on: March 14, 2020, 05:31:01 pm »
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Yes, my reply was written with that understanding. In my reply I was taking g(x) to be x^2 – 1 and f(x) = √x, and hence √(x^2 – 1) = f(g(x)).

Oh okay. Makes more sense now. Thanks a lot S_R_K. Much appreciated.  :)

Also, that method you used, can I use that for any question where the function is too difficult to draw like f(x)=√(x^2 – 1) when I have to find the domain and range?
« Last Edit: March 14, 2020, 05:37:45 pm by dream chaser »

S_R_K

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9624 on: March 14, 2020, 05:39:51 pm »
+1
Oh okay. Makes more sense now. Thanks a lot S_R_K. Much appreciated.  :)

Also, that method you used, can I use that for any question where the function is too difficult to draw like f(x)=√(x^2 – 1) when I have to find the domain and range?

Yes. You may still have to use calculus to find the range of the "outside" function, but at least you'll avoid an additional application of the chain rule and avoid more complicated equations to solve for stationary points.

dream chaser

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9625 on: March 14, 2020, 05:46:58 pm »
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Yes. You may still have to use calculus to find the range of the "outside" function, but at least you'll avoid an additional application of the chain rule and avoid more complicated equations to solve for stationary points.

Okay. Also, what do you mean about an "outside" function? What does that mean

S_R_K

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9626 on: March 14, 2020, 05:56:37 pm »
+1
Okay. Also, what do you mean about an "outside" function? What does that mean

It's just informal talk for composites. For a function which can be written as f(g(x)), we often call f(x) the "outside" function and g(x) the "inside" function.

dream chaser

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9627 on: March 14, 2020, 06:06:51 pm »
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It's just informal talk for composites. For a function which can be written as f(g(x)), we often call f(x) the "outside" function and g(x) the "inside" function.

Ok. Thanks for all the help S_R_K

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9628 on: March 27, 2020, 01:43:04 am »
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Hey guys

This might sound like a silly question, but is the question that I have attached beyond the scope of the course? :(

S_R_K

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Re: VCE Specialist 3/4 Question Thread!
« Reply #9629 on: March 27, 2020, 11:54:40 am »
+1
Hey guys

This might sound like a silly question, but is the question that I have attached beyond the scope of the course? :(

Answering that question doesn't require anything outside the study design, although I think it's natural to answer parts (b) and (c) (thus revealing the geometry of the situation) before answering part (a).