Bazza's 3/4 Question Thread

[pi]

Yeah, I said that earlier, cubics is usually the biggest they get in tech-free. Not sure why MQ overloads on it...

[WhoTookMyUsername]

thanks
another simple question

for a question such as

What rule or law prevents you from dividing by x? I know it's mathematically incorrect as x can = 0 but what specific mathematical law does it break (summarised in english) ? thanks

EDIT:

Also
2) is it most correct to say the answer is 0 or 3 or -2 or is 0,3,-2 sufficient?

[pi]

You lose a solution, that is what is mathematically incorrect (you are dividing by a variable). Dividing by a constant (say "3") is fine as there is no loss of solutions


edit: x=0 or x=3 or x=-2 or alternatively x={-2,0,3}

[paulsterio]

consider the case 0 * x = 0, which is true, no matter what value x takes, hence, you cannot have 0/0, it is undefined


i would personally just say x = 0, x = 3, x = 2

[dc302]

You lose a solution, that is what is mathematically incorrect (you are dividing by a variable). Dividing by a constant (say "3") is fine as there is no loss of solutions


edit: x=0 or x=3 or x=-2 or alternatively x={-2,0,3}

Where did you learn the last one? You can't say x is a set because it's a number, not a set.

edit: I think you meant x E {-2,0,3} ?

[TrueTears]

thanks
another simple question

for a question such as

What rule or law prevents you from dividing by x? I know it's mathematically incorrect as x can = 0 but what specific mathematical law does it break (summarised in english) ? thanks

like you said, you could be dividing by 0, you aren't breaking any mathematical laws, you're simply not abiding to axioms, hence if you did divide by x (http://en.wikipedia.org/wiki/Division_by_zero) the resultant may not be well-defined (http://en.wikipedia.org/wiki/Well-defined).

[dc302]

By first stating that x is not 0, and acknowledging that x=0 is a solution, you may then divide by x.

eg.

3x^3+3x^2-18x = 0

x=0 is a solution

Now suppose x=/=0, and we obtain 3x^2 + 3x - 18 = 0

Factorise further to get the remaining 2 solutions.

[totaled]

usually quadratics have xints, so what i do is just find the point in between the two x-intercepts
you probably would have done that in part a) or part b) of a question, and then in part c) or d) they ask you to find the tp, so just avg the two x-intercepts, and then sub that x coordinate int othe eqn to get your y point.

however, if they don't have x-intercepts (but i reckon 99% of the time in an exam/sac they will, cos they'll ask you to find them!), then yeah completing the square would have to do.

[WhoTookMyUsername]

thanks totaled


when writing range and domain what is standard way of expressing it?

in the textbook worked examples it goes

, and then and then


what is the standard / best way of expressing it?

2) Does R+ refer to real numbers including 0 and above?
is it best to avoid using this as "range"?

3) given a negative parabola with turning point at (1,9) (a=1) i am asked to find the equation.
What's the best way to look at this ? that ? is that the most common sense way ?

4) let's say for e.g. and the answers are , is it better to right the answer as x=3 AND x=1 or x= 3 OR x = 1? Is this the same for all similar equations?

5) for a question such as find the range given that the domain is [0,6] what is the best way to go about this type of question? Draw a graph?
thanks

[moekamo]

1) any way is good, you should use whichever you are most comfortable with, i normally say dom f = [-6, infinity], ran f=(5,10] or whatever it is, but really its just up to you.

2) R+ does NOT include 0 since 0 is not positive as R+ suggests. it's fine for expressing range, but if you think you'll get confused then avoid it.

3) yep that seems right, except you should have f(x)=-(x-1)^2 - 9, your one has its TP at (1,-9).

4) i think you'd say x=3 OR x=1, since then you can reject a solution if its out of the domain or whatever, but i could be wrong, i never thought too much about this...

[dc302]

5. Yeah drawing a graph would be easy. Mathematically, you would find the end points and the turning point (if it exists in the domain) and use this to find the max/min.

edit: and like moekamo said, a good way is domf=.... ranf=....
It is important to note that the domain and range are sets, so you should have them equal to a set. You can not say something like, ranf E [a,b], because this would suggest the range is a single element.

[pi]

You lose a solution, that is what is mathematically incorrect (you are dividing by a variable). Dividing by a constant (say "3") is fine as there is no loss of solutions


edit: x=0 or x=3 or x=-2 or alternatively x={-2,0,3}

Where did you learn the last one? You can't say x is a set because it's a number, not a set.

edit: I think you meant x E {-2,0,3} ?

ah, sorry. that's what i meant my bad

[paulsterio]

Personally, I always do it using x = -b/2a and then average the x-intercepts and check, 9 times out of 10, if they don't match, I've managed to catch an error whilst solving for the x-intercepts, usually bad factorising. x = -b/2a is really safe because it's easy to sub it, and yep, then just sub back in for y - completing the square is also something which I do.

It's hard to advise on what's the best method, because when you have enough experience, you intuitively know what to do because of recognition, like "oh, i'll complete the square here cause I can already see it" - kind of thing.

In regards to 1 - 5,
1. All are fine, I usually write - "ran f = ..." - I find that interval notation is easiest to use, but again, if the question uses set notation, I'd use set notation
3. If it's negative shouldn't a = -1? But yeah, that's how I would go about it too.
4. It doesn't really matter, I just put a comma: "x = 1, x = 2"
5. Drawing a graph would be best, but again, experience will give you intuition

This is how I would approach 5.

y = (x-4)^2 for x E [0,6]
This is a positive parabola with a turning point at x = 4 i.e. (4, 0). I know that 0 is smaller than 4 and 6 is greater than 4, so that means that the minimum y value occurs at x = 4, so it is y = 0
For the maximum, I know that the ends are x = 0 and x = 6, now with the turning point at x = 4, x = 0 is "further away" from the turning point, meaning it will be "higher" so maximum y will occur at x = 0, so y = 16. Since x = 0 is inclusive, this will be a square bracket.

The range is y E [0, 16]

[WhoTookMyUsername]

Thanks for all the help, just with the dom(f) = (5,9] so does the f refer to the f in f(x)? Would it be dom (g) if the equation is g(x)? If not what does f stand for? What if the equation is just a simple y=.. , just swy dom= ?

Thanks again

[TrueTears]

yeah it's just notation, dom f = dom f(x)