How to solve functional equations of different types.
In chapter functions one of the question which we all must be facing is how to solve functional equations for some given form.
Once we say functional equation , the immediate thoughts are what is functional equations ? How to solve function equation ? Are some of questions come to our mind.
Lets discuss what is functional equations ?
Many times we are not given the function rather a function is defined with respect to particular character of a function. Some times based on the relationship given and using tools of mathematics we can find the function.
e.g let f(x) be a function such that f(x+$\frac{1}{x})=\,x^2\,+\,\frac{1}{x^2}$
Then we can find f(x) as writing $$f(x+\frac{1}{x})=\,(x\,+\,\frac{1}{x})^2\,-\,2$$ $$ Thus\,\,\, f(x)=\,x^2\,-2$$
But many times given functional equation cannot be found e.g given f(x) is a function such that f(1) = 1 , f(2) = 4 , f(3) = 9 then it appears as if the function is f(x)=$x^2$ , But keep in mind just from three points on the curve we cannot find what this curve will be because we can have infinite functions satisfying above condition e.g $$f(x)=(x-1)(x-2)(x-3)g(x)\,+\,x^2$$
Where g(x) could be any function.
There are numerous such functional relations :
- f(x+y)=f(x)+f(y) for all x , y then one of function is f(x)=mx
- f(x.y) = f(x) + f(y) for all positive x , y one of function is f(x)=$log_a(x)$
- f(x+y) = f(x).f(y) for all x, y then on of the function is f(x)=$a^x$
In these above functional equations we can say about one of function but cannot say that they are the only function. To find exact solution of functional equation above relations alone are not sufficient we need more information about the functions.
Solution of functional Equation
There are certain specific character which we can express as functional equations like :-
- Functions symmetric about the line x = a and defined for all x to real is f(2a-x)=f(x) or can also be written as f(a-x)=f(a+x)
- Even function in particular symmetric about y axis function equation is f(-x)=f(x) for all x.
- Function symmetric about the point P(a,b) and defined for all x is f(2a-x)+f(x)=2b
- Odd functions symmetric about the origin will be f(-x)+f(x)=0 for all x.
Some of the spacial function whose functional equation can be defined are :
1. f(x) is a polynomial function satisfy the condition that $$f(x).f(\frac{1}{x})=k(f(x)+f(\frac{1}{x}))\,\,\,x\in R-{0}$$ find f(x)
$$Let\,\,\,f(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+…….+a_0\,\,a_n\neq0$$ $$\therefore \,f(\frac{1}{x})=\frac{a_n}{x^n}+\frac{a_{n-1}}{x^{n-1}}+\frac{a_{n-2}}{x^{n-2}}+………..+a_0 $$
As per given condition $f(x).f(\frac{1}{x})=k(f(x)+f(\frac{1}{x}))\,\,\,x\in R-{{0}}$
Comparing coefficient of $x^n$ on both sides we get $a_n.a_0=k.a_n$ $$ Hence\,\,\, a_0=k$$ Comparing coefficient of $x^{n-1}$ on both sides we get $$a_n.a_1+a_{n-1}.a_0=k.a_{n-1}$$ $$Since\,\,\, a_0=k\,\, \Rightarrow a_1=0$$ Comparing coefficient of $x^{n-2}$ on both sides we get $$a_n.a_2+a_{n-1}.a_1+a_{n-2}.a_0=k.a_{n-2}$$ $$Since\,\,\, a_0=k\,\, , a_1=0\,\Rightarrow a_2=0$$ Similarly all $a_r=0$ for all r= 1 , 2 , 3 ,….(n-1)
Now comparing constant terms on both sides we get $$ a_n^2+a_{n-1}^2+….+a_0^2=2.k^2$$ $$ \therefore a_n^2=k^2$$
Hence f(x)=k($x^n$+1) or f(x)=-k($x^n$-1)