Lol why are you waiting for me?y = -x^3
It's inverse is when you swap the x and y around so:
x = -y^3
Make y the subject:
y = cube root of -x
Now solving simultaneously:
-x^3 = cube root of -x
-x^9 = -x (Cube both sides)
x^9 = x
x = -1, 0, or 1
Will wait for Spiral or some one to confirm since I'm not 100% sure this is right.
Because I know you'll come in here and teach him everything from scratch lolLol why are you waiting for me?
The second example is not an inverse function unless you restrict it.The inverse function is drawn by flipping the line about the line y=x.
This means that when it intersects its inverse, it also intersects the line y=x.
So all you have to do is solve the pair of simultaneous equations y=-x^3 and y=x in order to find the inverse intercept.
I have provided a bit of a diagram to help you see it: View attachment 24028
Instead of solving simultaneously the equation of the curve and its inverse, you can simply do it with the equation of the curve, and the line y=x, since that's where it intersects the inverse anyway.
Here is another example for you to see: View attachment 24029
Yes I know it is not a function by definition due to the unrestricted domain, but it's so he can properly see the action of the 'flipping', and how it yields two solutions. If you restrict the domain beforehand, then you may miss out on solutions.The second example is not an inverse function unless you restrict it.
lol but this is the HSC. Asif they will expect that of a student.I think "find the intersection of a curve and its inverse" and "find the intersection of a function and its inverse function" are two pretty different questions... in particular the former is much nastier. try find the intersection of y = 10x^3 - 10x for instance and its inverse x = 10y^3 - y. You'll give yourself a serious headache (there are nine solutions!)
oooh thats legit... Id imagine you could cut it up and integrate it?lol but this is the HSC. Asif they will expect that of a student.
If you put the two curves you mentioned (with unrestricted domain) on a graphing program, you will see a 2x2 grid sorta thing. It would be a very interesting question to try to find the area of that '2x2' grid.
I've run across that theorem before when I randomly browse Maths articles on Wikipedia (I know it sounds sad, but it is actually quite interesting!). It would be interesting to have an elementary proof of it appear in the HSC.oooh thats legit... Id imagine you could cut it up and integrate it?
alsoooooo this reminds me, you know theres an awesome theorem about the intersection of two cubics? http://en.wikipedia.org/wiki/Cayley–Bacharach_theorem
lots of random geometric results are made almost trivial by that awesome theorem - you can treat things like a triple of lines, for instance, as a single cubic eqn e.g. (x+y-1)(x+2y-3)(x-4y+5) = 0 is a set of 3 lines, or a circle and a line e.g. (x^2+y^2-1)(x+y-1)=0. For example http://en.wikipedia.org/wiki/Pivot_theorem is a special case
reading maths articles on wikipedia is never a bad thing! I heard the guy who wrote the q8 this year based it on a theorem he found on wikipedia and I doubt theres an elementary proof of that theorem, Ive heard its really hard, and only works for cubics (has no quadratic/quartic analogue) which is weirdI've run across that theorem before when I randomly browse Maths articles on Wikipedia (I know it sounds sad, but it is actually quite interesting!). It would be interesting to have an elementary proof of it appear in the HSC.