Help with Questions

[ben_sebastian4]

Hey i'm pretty shit at maths so don't laugh if their easy to you guys.
Could you explain how to get the answer for these questions.

Question 9
The function  f : (–∞, a] → R with rule f (x) = x^3-3x^2+3  will have an inverse function provided.                                           
 
A. a ≤ 0
B. a ≥ 2
C. a ≥ 0
D. a ≤ 2[
E. a ≤ 1

Plus Q's 19, 20, and 22 from VCAA Exam 2 2010 in the multi. choice
http://www.vcaa.vic.edu.au/vce/studies/mathematics/cas/casexams.html

[b^3]

For the inverse function to exsits, you need to restrict the original function so that it is a one-to-one function. (i.e. it passes a horizontal line test). So for that function, you need to restrict it so that it continues to increase or decreasin (i.e. it does not start to turn back). So for this we are starting at -infinity, so we can go up to 0 (where it starts turning back a tthe turning point.

so the answer will be A. a is equal to or less than 0

[brightsky]

consider the general function f(x) =  x^3 - 3x^2 + 3
f'(x) = 3x^2 - 6x = 3x(x - 2) = 0
x = 0, x = 2
since it's the coefficient in front of x^3 is positive, hence the graph goes upwards
for there to be inverse function, it must be a one-to-one function, so we need to restrict the function to (-infty,0] or [0,2] or [2, infty) for an inverse function to exist.
clearly then with particular relevance to this question, A is the answer.

[ben_sebastian4]

Thanks, makes sense now

[brightsky]

as for your other questions:
19) the graph is a third degree function. hence we know that f(x) must be a fourth degree function. and since the f'(x) graph is positive, f(x) must be positive, hence only D will work.
20)
int^(a)_(0) f(x) dx = a
let x = u/5
dx/du = 1/5
so:
int^(u = 5a)_(u = 0) f(u/5) du = 5a
int^(5a)_(0) f(u/5) du + int^(5a)_(0) 3 du = 20a
multiply two to both sides to give 40a. so D.
22)
for this question, you basically need to resort to trial and error, although some obvious choices can be ruled out using a bit of intuition. E would be the correct answer since we you integrate it, you are left with a log, and we know that ln(ab) = ln(a) + ln(b).