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mathmath333
 one year ago
functions
mathmath333
 one year ago
functions

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mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} & \normalsize \text{Let }\ f(x) \text{be function such that } \hspace{.33em}\\~\\ & f(x)f(x+1)=f(x1)f(x2)f(x3)f(x4) ,\ \ x\geq 0 \hspace{.33em}\\~\\ & f(83)=81 \hspace{.33em}\\~\\ & f(77)=9 \hspace{.33em}\\~\\ & f(102)=? \hspace{.33em}\\~\\ & a.)\ 27 \hspace{.33em}\\~\\ & b.)\ 54 \hspace{.33em}\\~\\ & c.)\ 729 \hspace{.33em}\\~\\ & d.)\ \normalsize \text{Data insufficient} \hspace{.33em}\\~\\ \end{align}}\)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2it should have a fast way , the time given to solve such problem is average 2 min

dan815
 one year ago
Best ResponseYou've already chosen the best response.0that negative sign should really be there right, on the right side of the eqn

freckles
 one year ago
Best ResponseYou've already chosen the best response.3@mathmath333 I always like your questions. Can I ask where do you get them from?

dan815
 one year ago
Best ResponseYou've already chosen the best response.0here is something we do know dw:1435263889685:dw

dan815
 one year ago
Best ResponseYou've already chosen the best response.0and every multiplication of 2 of them can be rewritten recursively

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2i have just checked into solution set , it has given solution for \(f(81)\) and not \(f(102)\) as described in the question

dan815
 one year ago
Best ResponseYou've already chosen the best response.0f(77)=9 f(83)=9^2 f(102)=? and 9^3 is 729 if this turns up

dan815
 one year ago
Best ResponseYou've already chosen the best response.0it could be lol, i feel like the separation is too much cant write anythign recursively

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2lol i think the question is this book has a typo \(\large \color{black}{\begin{align} & \normalsize \text{Let }\ f(x) \text{be function such that } \hspace{.33em}\\~\\ & f(x)f(x+1)=f(x1)f(x2)f(x3)f(x4) ,\ \ x\geq 0 \hspace{.33em}\\~\\ & f(83)=81 \hspace{.33em}\\~\\ & f(77)=9 \hspace{.33em}\\~\\ &\color{red}{ f(81)}=? \hspace{.33em}\\~\\ & a.)\ 27 \hspace{.33em}\\~\\ & b.)\ 54 \hspace{.33em}\\~\\ & c.)\ 729 \hspace{.33em}\\~\\ & d.)\ \normalsize \text{Data insufficient} \hspace{.33em}\\~\\ \end{align}}\)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.227 is given as the answer.

dan815
 one year ago
Best ResponseYou've already chosen the best response.0okay lets see so 3^2, 3^3 and 3^4 for f(77), f(77+4), f(77+6)

freckles
 one year ago
Best ResponseYou've already chosen the best response.3actually I think we get \[(f(81))^2=729\]

dan815
 one year ago
Best ResponseYou've already chosen the best response.0can we do another question which we know doesnt have a typo xD

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} & f(82)f(83)=f(81)f(80)f(79)f(78) \ \color{red}{(1)}\hspace{.33em}\\~\\ & f(81)f(82)=f(80)f(79)f(78)f(77) \ \color{red}{(2)}\hspace{.33em}\\~\\ \end{align}}\) dividing 1 and 2 works

freckles
 one year ago
Best ResponseYou've already chosen the best response.3\[f(82)f(83)=f(81)f(80)f(79)f(78) \\ f(82) \cdot 81 =f(81)f(80)f(79)f(78) \\ 81=\frac{f(81)f(80)f(79)f(78)}{f(82)} \\ \text{ now we plug \in } 81 \\ f(81)f(82)=f(80)f(79)f(78)f(77) \\ f(81)=\frac{f(80)f(79)f(78)}{f(82)} f(77) \\ \text{ multiply both sides by} f(81) \\ f(81) \cdot f(81)=\frac{f(81)f(80)f(79)f(78)}{f(82)} f(77)\]

dan815
 one year ago
Best ResponseYou've already chosen the best response.0thats really cool, give me another one!

freckles
 one year ago
Best ResponseYou've already chosen the best response.3we already said that one thingy was 81 so you have \[(f(81))^2=81 f(77)\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.3and I agree with @dan815 more please!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I better go to sleep! very clever @freckles :)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2well i only post the one's i stuck at, let's see if i have some tough ones

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} & \normalsize \text{let}\ f(x)=121x^2,\ g(x)=x8+x+8 \hspace{.33em}\\~\\ & \normalsize \text{and}\ h(x)=\text{min}(f(x),g(x)). \hspace{.33em}\\~\\ & \normalsize \text{What is the number of integer values of x for which } \hspace{.33em}\\~\\ & h(x)\ \normalsize \text{is equal to a positive integral value? } \hspace{.33em}\\~\\ & a.)\ 17 \hspace{.33em}\\~\\ & b.)\ 19 \hspace{.33em}\\~\\ & c.)\ 21 \hspace{.33em}\\~\\ & d.)\ 23 \hspace{.33em}\\~\\ \end{align}}\)

freckles
 one year ago
Best ResponseYou've already chosen the best response.3ok I'm here and looking now

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1by induction, I got this: \[\Large {\left\{ {f\left( n \right)} \right\}^2} = f\left( {n + 2} \right) \times f\left( {n  4} \right)\]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1setting n=81, we get: \[\Large {\left\{ {f\left( {81} \right)} \right\}^2} = f\left( {83} \right)f\left( {77} \right)\] namely the result of @freckles

freckles
 one year ago
Best ResponseYou've already chosen the best response.3I have to come back and look this one crab time sorry

dan815
 one year ago
Best ResponseYou've already chosen the best response.0i dont quite understand what does it mean min(f(x),g(x))

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2u need to graph that f(x) and g(x) and find the intersection , and as it asks minimum u have to choose the lowest part with respect to that

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2it can also be done without graphing

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2example min(f(x),g(x)) for dw:1435266966304:dw f(x)=x,g(x) =x^2+5

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2dw:1435267032841:dw

dan815
 one year ago
Best ResponseYou've already chosen the best response.0i think to start off look at g1(x) = x8 and g2(x) = x+8

dan815
 one year ago
Best ResponseYou've already chosen the best response.0this will give us a constant when 0<x < 8

dan815
 one year ago
Best ResponseYou've already chosen the best response.0it will be 16 in that domain

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2it looks line thisdw:1435267447526:dw

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2dw:1435267567368:dw

dan815
 one year ago
Best ResponseYou've already chosen the best response.0u want the number of integers of x in that intersection?

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2dw:1435267618964:dw

dan815
 one year ago
Best ResponseYou've already chosen the best response.0between the intersection?

dan815
 one year ago
Best ResponseYou've already chosen the best response.0or just the min on the abs value?

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2well i m still confused on interpreeting the quetion let me think

dan815
 one year ago
Best ResponseYou've already chosen the best response.0ya me too the min(f(x),g(x)) still dunno what that means exacttllyy

dan815
 one year ago
Best ResponseYou've already chosen the best response.0there are 17 points from that flat line, for hte integers then we have y=2x and y=2x lines from the abs value

dan815
 one year ago
Best ResponseYou've already chosen the best response.0yeah there will be 2 more from the intersection on each side

dan815
 one year ago
Best ResponseYou've already chosen the best response.0we see 11^2x^2=2x x^2+2x11^2=0 solve for the roots and see how many integers we can fit

dan815
 one year ago
Best ResponseYou've already chosen the best response.0you get sqrt 122 1 so its greater than 10 barely that means another 2 from the right side and another 2 from the left side

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2it is not giving rational roots hmm

dan815
 one year ago
Best ResponseYou've already chosen the best response.0we just want to see the bound

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2yea we need to count the number line of x of the intersection part.

dan815
 one year ago
Best ResponseYou've already chosen the best response.0tbh i think the question is worded wrong it should just say.. how many integer solutions are there that are less than the quadratic

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2these are framed to confuse students majority are confuzing

dan815
 one year ago
Best ResponseYou've already chosen the best response.0lol thats annoying . they should confuse us with hard questions, not easy ones worded badly

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2my head spins now going to sleep

dan815
 one year ago
Best ResponseYou've already chosen the best response.0okay cya! thanks for the questions

dan815
 one year ago
Best ResponseYou've already chosen the best response.0looks like you already found this but ill leave this here, just in case on how we got the equations 2x and 2x for the lines

freckles
 one year ago
Best ResponseYou've already chosen the best response.3there are 21 integers between 11 and 11 (not including the endpoints) because at the endpoints h=0 which isn't a positive number

freckles
 one year ago
Best ResponseYou've already chosen the best response.3like i didn't want to include the endpoints because I want h positive

freckles
 one year ago
Best ResponseYou've already chosen the best response.3and anything outside the interval I mentioned h would be negative

freckles
 one year ago
Best ResponseYou've already chosen the best response.3dw:1435269002394:dw

freckles
 one year ago
Best ResponseYou've already chosen the best response.3for example h(10)=g(10) (since f(10)>g(10)) ... h(5)=g(5) (since f((5)>g(5)) ... h(0)=g(0) (since f(0)>g(0)) ..so on... h(x)=g(x) for integer solutions between 10 and 10 (inclusive) h(x)=f(x) for integers solutions on (inf,11] union [11,inf) but h(x)<=0 there so what we want to look at is: h(x)=g(x) for integer solutions between 10 and 10 (inclusive) since h(x)>0 here \[x8+x+8=121x^2 \\ \text{ from our graph one solution occurs on } (\inf,8) \\ (x8)(x+8)=121x^2 \\ (x1)^2=122 \\ x= 1 \pm \sqrt{122} \\ x=1\sqrt{122} \text{ is only valid on that interval } \\ x \approx 10.045 \\ x8+x+8=121x^2 \\ \text{ the other solution \to this occurs on } (8,\infty) \\ x8+x+8=121x^2 \\ (x+1)^2=122 \ x=1 \pm \sqrt{121} \\ x=1 +\sqrt{122} \text{ is only valid on that interval } \\ x=10.045\] dw:1435269711175:dw

freckles
 one year ago
Best ResponseYou've already chosen the best response.3anyways just saying all of this just in case you guys were still confused on the h=min(f,g) thing
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