VCE Methods Question Thread!

[Sine]

Hey guys,

Could i please have some clarification with the following question(s), working out by hand would be much appreciated, im pretty sure i have the right answers just want to double check

Q1: Consider the function defined by f(x) = 2(x-3)^2.
a)   Find the rule for the inverse.
b)   Restrict the domain of f to the form of [a, infinity] so that the inverse is also a function.
c)   State the rules for the restricted f and f^-1 using function notation.
d)   Show that f(f^-1(x)) = x

Q2: For: mx +2y = n and 3x + 6y = -1, find m and n for which the equations have:
a) a unique solution
b) an infinite number of solutions
b) no solution

I am happy to look over your solutions to see if they are correct

[sara.vce]

Q1: Consider the function defined by f(x) = 2(x-3)^2.
a)   Find the rule for the inverse.
b)   Restrict the domain of f to the form of [a, infinity] so that the inverse is also a function.
c)   State the rules for the restricted f and f^-1 using function notation.
d)   Show that f(f^-1(x)) = x

Here's what I would do for Question 1, hope it helps!

Q1:
a)
swap x and y
x = 2(y-3)^2
x/2 = (y-3)^2
y = sqrt(x/2)+3
b)
domain of f(x) must be one:one therefore
[x-value of TP, infinity)
[3, infinity)
c)
f: [3, infinity)--> R, f(x) = 2(x-3)^2
f^-1: [0, infinity) --> R, f^-1(x)= sqrt(x/2) + 3
d)
f(sqrt(x/2) + 3)
so f(f^-1(x)) = 2(x-3)^2 where x = sqrt(x/2) + 3
2(sqrt(x/2)+3-3)^2
2(x/2)
= x

[LachlanBarr8]

Thank you!

[Sine]

Here's what I would do for Question 1, hope it helps!

Q1:
a)
swap x and y
x = 2(y-3)^2
x/2 = (y-3)^2
y = sqrt(x/2)+3
b)
domain of f(x) must be one:one therefore
[x-value of TP, infinity)
[3, infinity)
c)
f: [3, infinity)--> R, f(x) = 2(x-3)^2
f^-1: [0, infinity) --> R, f^-1(x)= sqrt(x/2) + 3
d)
f(sqrt(x/2) + 3)
so f(f^-1(x)) = 2(x-3)^2 where x = sqrt(x/2) + 3
2(sqrt(x/2)+3-3)^2
2(x/2)
= x

Alrighty,

Q1(a) You firstly need to define y= then state swap 'x" and "y"
You also need justification on why you decide taking the positive root (this will depend on what the domain is for the question)
Also to finish your question "y" is not notation for the inverse of a function. You need to state f^-1(x) =  in order to get full marks

(b) Your answer should be fine

(c) Good

(d) For show that questions it is easy for the assessor if you take a systematic approach.
E.g. state what is meant to be proven
Then go LHS = etc + etc1 = etc2 = RHS
Thus LHS = RHS

[Azila2004]

Hello guys!

I'm struggling with this extended response question, which states the following:

AB and CD are crossroads. A jogger runs along AB at 8km/hr and passes O, the intersection, at 1pm. A second jogger runs along CD at 10km/hr and passes O at 1:30 pm. (Imagine two perpendicular lines just intersecting each other). Let y be the distance between them and t be the number of hours after 1pm. Find an expression for y in terms of t.

I feel bad asking you guys, but the answers section of the Cambridge book does not explain it.

Thank you!

[fun_jirachi]

Hey there!

The jogger on AB passes O at 1pm, so at any point, their displacement from O in terms of hours past 1pm will just be their velocity multiplied by the time differece - ie. 8t.

The jogger on CD is travelling at a velocity of 10kmh^-1 and is half an hour away from O at 1pm - ie. they are 5km away at 1pm, travelling towards O, then passing it afterwards. Basically we have that they are '5km behind' - we use the same logic for the jogger on AB now to denote the displacement as 10t-5.

Then, we have that the jogger's displacements forming a right angled triangle with a right angle at O, so we have the distance \(y = \sqrt{(10t-5)^2 + (8t)^2} = \sqrt{(164t^2 - 100t +25)}\).

Hope this helps

[Azila2004]

Thanks so much!

[Natnael Gizaw]

Hey guys,

I have a question that I need help with involving matrices and sequence of transformations

Let f(x) = 3x − 2 and g(x) = x^2 − 4x + 2.
What is the sequence of transformations that takes the
graph of y = g(x) to the graph of y = g(f(x)) ?

Thanks

[S_R_K]

Hey guys,

I have a question that I need help with involving matrices and sequence of transformations

Let f(x) = 3x − 2 and g(x) = x^2 − 4x + 2.
What is the sequence of transformations that takes the
graph of y = g(x) to the graph of y = g(f(x)) ?

Thanks

Write both g(x) and g(f(x)) in the form a(x – h)^2 + k, and then find the sequence of transformations either by inspection or using the mapping method.

[Rose34]

Could someone please help me with this question it is from methods units 1&2

Find the domain and range?

y= 2/(2x-5)^2

Thanks in advance

[SmartWorker]

Could someone please help me with this question it is from methods units 1&2

Find the domain and range?

y= 2/(2x-5)^2

Thanks in advance

Try graphing (use desmos) and have a look. This is a truncus as it has the general form of 1/x^2 so the graph is above x-axis. (for this case)

Otherwise work out the asymptotes of the graph.

Let 2x-5 =0. Thus,  asymptote is at X = 5/2., since x cannot take a value whereby the denominator becomes 0 (undefined) Since the graph is above x-axis, y asymptote is y=0
 
domain --> What values does x take
range --> What values does y take
Domain: R/{5/2} ---> means all real numbers except when x =5/2
Range: (0, positive infinity) since y = 0 is an asymptote.

[Geoo]

My cas (casio classpad) won't solve a surd.. It will simplify them, but it won't solve something as basic as the square root of 5.

Could someone tell my the step by step?

I feel so stupid for asking this...

[whys]

My cas (casio classpad) won't solve a surd.. It will simplify them, but it won't solve something as basic as the square root of 5.

Could someone tell my the step by step?

I feel so stupid for asking this...

At the bottom, it says 'standard'. Click on that, and change it to decimal.

[ArtyDreams]

My cas (casio classpad) won't solve a surd.. It will simplify them, but it won't solve something as basic as the square root of 5.

Could someone tell my the step by step?

I feel so stupid for asking this...

Don't feel like that! It can def take a while to adjust to a new CAS.

If you need to find the square root of 5, have you ensured that your CAS is in decimal mode? It should give you a decimal answer then when you type in root5

[Geoo]

Don't feel like that! It can def take a while to adjust to a new CAS.

If you need to find the square root of 5, have you ensured that your CAS is in decimal mode? It should give you a decimal answer then when you type in root5

At the bottom, it says 'standard'. Click on that, and change it to decimal.

Thanks guys, yeah it was on standard.....
I was trying everything but that...