Question

functional equations

Answer 1

love em or leave em

Answer 2

\[\color{brown}{f(x)+(x+1)^3=2f(x+1)}\]

Answer 3

Find \(\color{#C00}{f(10)}\)

Answer 4

lol thanks for posting the full question @ParthKohli

Answer 5

\[f(-1) = 2f(0)\]

Answer 6

f(0)+1=2f(1)

Answer 7

wat to do next i 3 variables 2 equations??

Answer 8

\[f(0) = 2f(1) - 1 \\ f(1) = 2f(2) - 8\]So\[f(0) = 2(2f(2) - 8) = 4f(2) - 16\]

Answer 9

So hard...

Answer 10

\[f(0)=4(f(2)-8)-1=4f(2)-16-1=4f(2)-17\]

Answer 11

is f always of the from\[ax^2+bx+c\]

Answer 12

Ah my bad, yes.

Answer 13

No, it's a polynomial. It can be in that form, but we're not sure.

Answer 14

so far we see \[f(10)+11^3=2f(11)\]
\[f(9)+10^3=2f(10)\]
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\[f(10)=f(9)-2f(11)+11^3-10^3\]

Answer 15

I think that it must be in the form \(ax^3 + bx^2 + cx + d\)

Answer 16

\[f(x-1)+x^3=2f(x)\]

Answer 17

we need a relation between f(9) and f(11)

Answer 18

\[f(9) = 2f(10) - 1000\]\[f(10) = 2f(11) - 1331\]\[\iff f(9) = 2(f(11) - 1331 ) - 1000\]

Answer 19

\[f(9) = 2f(11) - 2662 - 1000\]

Answer 20

That's the relation.

Answer 21

isnt this the solution then

Answer 22

since
\[f(9)-2f(11)=3662\]
\[f(10)=3662+11^3-10^3\]

Answer 23

OMG!

Answer 24

But it's an integer between 0 and 999.

Answer 25

okay so we found a relationship between numbers 2 units away from each other x-1 and x+1
\[\color{blue}{f(x-1)+x^3=2f(x).......... f(x)+(1+x)^3=2f(1+x)}\]
\[f(x-1)+x^3=4f(x+1)+2(x+1)^3\]

Answer 26

\[f(9)-4f(11)=2(11)^3-10^3\]

Answer 27

earlier we used 2 not 4

Answer 28

Ahhh.

Answer 29

@Hero

Answer 30

Let f(x)=a+bx+cx^2+dx^3
NOTE: f(x) must be a third degree polynomial

Answer 31

Now, f(x)+(x+1)^3 =2f(x+1)

Answer 32

\[\huge f(10)=f(9)-2f(11)+11^3-10^3=???\]

Answer 33

use it to find a,b,c and d

Answer 34

I see, yeah!

Answer 35

u will get
a=-13
b=9
c=-3
d=1

Answer 36

Equating coefficients

Answer 37

yep

Answer 38

Yay

Answer 39

now, once u know f(x)
u can find f(10)

Answer 40

help me i dont get where you get the coeeffients from

Answer 41

@Jonask a technique.\[ax + by = 2x + 3y\]means\[a=2,b=3\]

Answer 42

oh great

Answer 43

f(x)+(x+1)^3 =2f(x+1)
a+bx+cx^2+dx^3 +x^3+3x^2+3x+1 = 2{a+b(x+1) +c(x+1)^2 +d(x+1)^3

Answer 44

can u continue from here?

Answer 45

\[f(10) = 10^3 - 3(10)^2 +9(10) -13 = 1000 - 300 + 90 - 13 \\ = 777 \]

Answer 46

thats correct

Answer 47

\[\huge \color{green}{THANKS}\]

Answer 48

welcome