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inv functions question from 2004 3U HSC paper (1 Viewer)

gamja

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Q5b of the above hsc paper:
- The graphs of y=f(x) [1672655424016.png for x>=0] and y=invf(x) [which I calculated to be 1672655519062.png]meet at exactly one point P. Let α be the x-coordinate of P. Explain why α is a root of the equation

x^3 +x -1 = 0.

I let both function formulae equal each other and squared them and eventually came out with a long 5-th degree polynomial. Any ideas? (much appreciated :)
 

Nedom

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Don't equations and their inverse intercept at y=x. (I don't even know if that's relevant, just something to consider). Then you could just find the root of the equation. Then equate, huzzah.
 

Nedom

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Wait lmao, it's so ez. I am so stupid. If you equate the f(x) and y =x, you rearrange you get the equation, and that gives the answer. LMAO, too rusty at maths.
(So same thing as I said before, but you don't need to solve for the root of the equation, it's unneccessary)
(Didn't actually try it so my first response is half-assed, don't worry about that one)
 

gamja

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Wait lmao, it's so ez. I am so stupid. If you equate the f(x) and y =x, you rearrange you get the equation, and that gives the answer. LMAO, too rusty at maths.
(So same thing as I said before, but you don't need to solve for the root of the equation, it's unneccessary)
(Didn't actually try it so my first response is half-assed, don't worry about that one)
very true [highkey embarrassing that i forgot the y=x rule omg] thank you
 

Average Boreduser

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Q5b of the above hsc paper:
- The graphs of y=f(x) [View attachment 37328 for x>=0] and y=invf(x) [which I calculated to be View attachment 37329]meet at exactly one point P. Let α be the x-coordinate of P. Explain why α is a root of the equation

x^3 +x -1 = 0.

I let both function formulae equal each other and squared them and eventually came out with a long 5-th degree polynomial. Any ideas? (much appreciated :)
Bro why u doing 2004 3u questions lmfao that literally 2 decades old
 

gamja

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Bro why u doing 2004 3u questions lmfao that literally 2 decades old
just going thru 3u revision across entire syllabus starting from FWWF lol probably the last opportunity i have to do that as well
 

Trebla

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Bro why u doing 2004 3u questions lmfao that literally 2 decades old
FYI the 2022 Maths Ext2 HSC had a question that was almost identical to one in the 2005 Maths Ext2 HSC. Just sayin…
 

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