functions

- mathmath333

functions

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- mathmath333

\(\large \color{black}{\begin{align}
& \normalsize \text{Let }\ f(x) \text{be function such that } \hspace{.33em}\\~\\
& f(x)f(x+1)=-f(x-1)f(x-2)f(x-3)f(x-4) ,\ \ x\geq 0 \hspace{.33em}\\~\\
& f(83)=81 \hspace{.33em}\\~\\
& f(77)=9 \hspace{.33em}\\~\\
& f(102)=? \hspace{.33em}\\~\\
& a.)\ 27 \hspace{.33em}\\~\\
& b.)\ 54 \hspace{.33em}\\~\\
& c.)\ 729 \hspace{.33em}\\~\\
& d.)\ \normalsize \text{Data insufficient} \hspace{.33em}\\~\\
\end{align}}\)

- mathmath333

it should have a fast way , the time given to solve such problem is average 2 min

- dan815

that negative sign should really be there right, on the right side of the eqn

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## More answers

- mathmath333

yes its there

- dan815

is it C?

- dan815

orr Data insufficient

- freckles

@mathmath333 I always like your questions. Can I ask where do you get them from?

- dan815

here is something we do know |dw:1435263889685:dw|

- mathmath333

http://www.flipkart.com/quantitative-aptitude-quantum-cat-common-admission-test-into-iims-english/p/itmdygg8z2ggtuk2?pid=9789351416401&ppid=9789351416401

- dan815

and every multiplication of 2 of them can be rewritten recursively

- mathmath333

i have just checked into solution set , it has given solution for \(f(81)\) and not \(f(102)\) as described in the question

- dan815

f(77)=9
f(83)=9^2
f(102)=?
and 9^3 is 729 if this turns up

- dan815

it could be lol, i feel like the separation is too much cant write anythign recursively

- mathmath333

lol i think the question is this book has a typo
\(\large \color{black}{\begin{align}
& \normalsize \text{Let }\ f(x) \text{be function such that } \hspace{.33em}\\~\\
& f(x)f(x+1)=-f(x-1)f(x-2)f(x-3)f(x-4) ,\ \ x\geq 0 \hspace{.33em}\\~\\
& f(83)=81 \hspace{.33em}\\~\\
& f(77)=9 \hspace{.33em}\\~\\
&\color{red}{ f(81)}=? \hspace{.33em}\\~\\
& a.)\ 27 \hspace{.33em}\\~\\
& b.)\ 54 \hspace{.33em}\\~\\
& c.)\ 729 \hspace{.33em}\\~\\
& d.)\ \normalsize \text{Data insufficient} \hspace{.33em}\\~\\
\end{align}}\)

- mathmath333

27 is given as the answer.

- dan815

okay lets see so
3^2, 3^3 and 3^4
for
f(77), f(77+4), f(77+6)

- freckles

actually I think we get
\[(f(81))^2=729\]

- freckles

let me post my work

- dan815

can we do another question which we know doesnt have a typo xD

- mathmath333

\(\large \color{black}{\begin{align}
& f(82)f(83)=-f(81)f(80)f(79)f(78) \ -\color{red}{(1)}\hspace{.33em}\\~\\
& f(81)f(82)=-f(80)f(79)f(78)f(77) \ -\color{red}{(2)}\hspace{.33em}\\~\\
\end{align}}\)
dividing 1 and 2 works

- dan815

oh beauty

- freckles

\[f(82)f(83)=-f(81)f(80)f(79)f(78) \\ f(82) \cdot 81 =-f(81)f(80)f(79)f(78) \\ 81=\frac{-f(81)f(80)f(79)f(78)}{f(82)} \\ \text{ now we plug \in } 81 \\ f(81)f(82)=-f(80)f(79)f(78)f(77) \\ f(81)=\frac{-f(80)f(79)f(78)}{f(82)} f(77) \\ \text{ multiply both sides by} f(81) \\ f(81) \cdot f(81)=\frac{-f(81)f(80)f(79)f(78)}{f(82)} f(77)\]

- dan815

thats really cool, give me another one!

- freckles

we already said that one thingy was 81
so you have
\[(f(81))^2=81 f(77)\]

- freckles

and I agree with @dan815 more please!

- anonymous

I better go to sleep! very clever @freckles :-)

- mathmath333

well i only post the one's i stuck at, let's see if i have some tough ones

- mathmath333

\(\large \color{black}{\begin{align}
& \normalsize \text{let}\ f(x)=121-x^2,\ g(x)=|x-8|+|x+8| \hspace{.33em}\\~\\
& \normalsize \text{and}\ h(x)=\text{min}(f(x),g(x)). \hspace{.33em}\\~\\
& \normalsize \text{What is the number of integer values of x for which } \hspace{.33em}\\~\\
& h(x)\ \normalsize \text{is equal to a positive integral value? } \hspace{.33em}\\~\\
& a.)\ 17 \hspace{.33em}\\~\\
& b.)\ 19 \hspace{.33em}\\~\\
& c.)\ 21 \hspace{.33em}\\~\\
& d.)\ 23 \hspace{.33em}\\~\\
\end{align}}\)

- mathmath333

@dan815 @freckles

- freckles

ok I'm here and looking now

- mathmath333

yep good luck

- Michele_Laino

by 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

setting 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

I have to come back and look this one
crab time
sorry

- dan815

i dont quite understand what does it mean
min(f(x),g(x))

- mathmath333

u 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

it can also be done without graphing

- dan815

oh ok i see now

- mathmath333

example min(f(x),g(x)) for |dw:1435266966304:dw| f(x)=-x,g(x) =-x^2+5

- mathmath333

|dw:1435267032841:dw|

- dan815

i think to start off look at g1(x) = |x-8|
and g2(x) = |x+8|

- dan815

this will give us a constant when 0<|x| < 8

- dan815

it will be 16 in that domain

- dan815

from -8 to 8

- mathmath333

it looks line this|dw:1435267447526:dw|

- dan815

yes exactly

- dan815

|dw:1435267527733:dw|

- mathmath333

|dw:1435267567368:dw|

- dan815

u want the number of integers of x in that intersection?

- mathmath333

|dw:1435267618964:dw|

- dan815

between the intersection?

- dan815

|dw:1435267649589:dw|

- dan815

or just the min on the abs value?

- mathmath333

well i m still confused on interpreeting the quetion let me think

- dan815

ya me too the min(f(x),g(x)) still dunno what that means exacttllyy

- dan815

there are 17 points from that flat line, for hte integers then we have y=2x and y=-2x lines from the abs value

- mathmath333

anser given is 21

- dan815

yeah there will be 2 more from the intersection on each side

- dan815

we see
11^2-x^2=2x
x^2+2x-11^2=0
solve for the roots and see how many integers we can fit

- dan815

you 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

- dan815

17+4=21

- mathmath333

it is not giving rational roots hmm

- dan815

we just want to see the bound

- mathmath333

yea we need to count the number line of x of the intersection part.

- dan815

tbh i think the question is worded wrong it should just say.. how many integer solutions are there that are less than the quadratic

- mathmath333

these are framed to confuse students majority are confuzing

- dan815

lol thats annoying -.- they should confuse us with hard questions, not easy ones worded badly

- mathmath333

my head spins now going to sleep

- dan815

okay cya! thanks for the questions

- mathmath333

thnks

- dan815

looks 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

- dan815

|dw:1435268655727:dw|

- dan815

|dw:1435268725528:dw|

- freckles

there are 21 integers between -11 and 11 (not including the endpoints)
because at the endpoints h=0 which isn't a positive number

- freckles

like i didn't want to include the endpoints because I want h positive

- freckles

and anything outside the interval I mentioned h would be negative

- freckles

|dw:1435269002394:dw|

- freckles

for 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
\[|x-8|+|x+8|=121-x^2 \\ \text{ from our graph one solution occurs on } (-\inf,-8) \\ -(x-8)-(x+8)=121-x^2 \\ (x-1)^2=122 \\ x= 1 \pm \sqrt{122} \\ x=1-\sqrt{122} \text{ is only valid on that interval } \\ x \approx -10.045 \\ |x-8|+|x+8|=121-x^2 \\ \text{ the other solution \to this occurs on } (8,\infty) \\ x-8+x+8=121-x^2 \\ (x+1)^2=122 \ x=-1 \pm \sqrt{121} \\ x=-1 +\sqrt{122} \text{ is only valid on that interval } \\ x=10.045\]
|dw:1435269711175:dw|

- freckles

anyways just saying all of this just in case you guys were still confused on the h=min(f,g) thing

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