Question

If 4f(x) + f(3-x) = x^2 what is f(x)? Can somebody guide me through this?

Answer 1

Subtract f(3-x) from both sides and then divide both sides by 4.

Answer 2

hmmm

Answer 3

so... it would be f(x)= \[x^{2}-(3-x)/4\]

Answer 4

i think the gimmick is to replace x by 3 - x in the first equation and get
\[4f(3-x)+f(x)=(3-x)^2\] and then you have two equations
\[4f(x)+f(3-x)=x^2\]
\[4f(3-x)+f(x)=(3-x)^2\] then some algebra to solve for
\[f(x)\] if i recall a system of two equations. let me write it out on paper but i am pretty sure this is the method you need

Answer 5

wow. that helps a lot. I wasnt sure if i could just subtract 3-x. It was part of the function.

Answer 6

Multiply the first equation by four
Minus the first one by the second one
Isolate f(x)

Answer 7

it is cumbersome to write, so lets put
\[a=f(x),b=f(3-x)\] and solve
\[4a+b=x^2\]
\[4b+a=x^2-6x+9\] or the system
\[4a+b=x^2\]
\[a+4b=x^2-6x+9\] we want "a" so lets multiply the first equation by 4 and get
\[8a+4b=4x^2\] subtract the second equation from it and get
\[3a=3x^2+6x-9\] divide by 3 and get
\[a=x^2+2x-3\] so
\[f(x)=x^2+2x-3\] check my algebra

Answer 8

well that was wrong. when i subtract i should have
\[7a=3x^2+6x-9\] making
\[a=\frac{3x^2+6x-9}{7}\]
first one i made a mistake

Answer 9

But if a = f(x), it doesn't check in the original equation.

Answer 10

Actually it should be a =( x^2+2x-3)/5

Answer 11

so f(x) = (x^2 + 2x -3) /5?

Answer 12

Yes. Check is out by using various x values.

Answer 13

where did the 5 come from?

Answer 14

-15 f(x) = -4x^2+x^2 -6x+9

Answer 15

That was the result after eliminating 4f(3-x) using elimination.

Answer 16

oh of course! i is true that
\[4\times 4=16\] not for example 8 like i wrote

Answer 17

so we should have
\[15a=15f(x)=3x^2+6x-9\] and so
\[f(x)=\frac{x^2+2x-3}{5}\] damn that arithmetic

Answer 18

Ain't it the truth!!

Answer 19

and the check is good!
http://www.wolframalpha.com/input/?i=4%28x^2%2B2x-3%29%2F5%2B%28%283-x%29^2%2B2%283-x%29-3%29%2F5%29

Answer 20

Hey guys, thanks so much. Can you help me determine the total revenue function of another problem?