**Math Trick-7**

*Functional Equations is a chapter which is essential for Olympiad,ISI,CMI,IOMA entrances.*

*An equation involving an unknown function is called a*

__functional equation.__**Ex**. Considering Cauchy's functional equation:-

**f(x+y)=f(x) + f(y)**;

Generally while you are solving a problem of functional equation with two variables x & y then in most of the situations to solve a given problem you are to put these values

Put x= -x

Put y= -x

Put x=f(y)

Put x=y

Put x=f(x)

etc . . .

The choices differ & it completely depends on the given problem which requires these selections. So, just use your brain & start solving problem. I am giving you some examples.

**Ex.1. [ISI B.Math sample paper] If f(x+y)=f(x)+f(y) for all y in R. Then f(x) is a function which is**

**A. Odd B.Even C. None**

Sol:-(A) Putting x= -x ,then

f(x)+f(-x)=f(0) -----(1)

Again put x=0 in (1), so f(0)=0.

So, from (1), f(-x)= -f(x) which is an odd function.

**Ex.2. [RMO paper] If the function f satisfies the relation**

**f(x+y)+f(x-y)=2f(x).f(y) for all x,y in R.**

**And f(0)≠0. Prove that f(x) is even function.**

Sol:- f(x+y)+f(x-y)=2f(x).f(y) ---(1)

Replace x by y & y by x

Then f(y+x)+f(y-x)=2f(y)f(x) ---(2)

Solving (1) & (2), we get

f(y-x)=f(x-y)

Put y=2x then f(x)=f(-x).

Hence f(x) is an even function.

**Ex.3.[RMO paper] Determine all**

**functions f:R->R such that**

**f(x-f(y))=f(f(y))+xf(y)+f(x)-1, x,y€R.**

Sol:- Put x=f(y)=0

Then f(0)=1

Again put x=f(y)=k,then

f(0)=2f(k)-1+(k^2)

=> f(k)= 1- (k^2)/2 .

Hence, f(x)=1- (x^2)/2 is the unique equation.

**Ex.4.[ISI MMA SAMPLE PAPER] If f(x) is a real**

**valued fiction such that**

**2f(x) + 3f(-x) = 15 - 4x , for all x.**

**Then f(2) is**

**A.-15 B.22 C.11 D.0**

Sol:- (C) Given 2f(x) + 3f(-x) = 15 - 4x -----(1)

put x = -x in the given functional equation (1),

we have 2f(-x) + 3f(x) = 15 + 4x ------------(2)

Solving (1) & (2) we have f(x) = 3 + 4x .

f(2)=11.