Functional Equations
Functional equation is any equation that specifies a function in implicit form. Often, the equation relates the value of a function (or functions) at some point with its values at other points. For instance, the properties of functions can be determined by considering the types of functional equations the satisfy.
Functional equations satisfied by typical Functions:
1. f (x + y) = f(x).f(y) is satisfied by f(x) = ax as f (x + y) = ax+y = ax ay = f(x) f(y).
2. f (x + y) = f (x – y) = f(x)/f(y) is satisfied by f (x) = ax as f (x -y) = ax-y \(=\frac{{{a}^{x}}}{{{a}^{y}}}=\frac{f(x)}{f(y)}\),
3. f(x) + f(y) = f(x y) is satisfied by f(x) = as f(x) + f(y) = logax + logab
logaxy = f (xy)
4. f(x) – f(y) = f(x/y) is satisfied by f(x) = logex as f(x) – f(y) = logex – logey \(={{\log }_{e}}\left( \frac{x}{y} \right)=f\left( \frac{x}{y} \right)\),
5. f(x) ± f(y) = \(f\left( x\sqrt{1-{{y}^{2}}}\pm y\sqrt{1-{{x}^{2}}} \right)\) is satisfied by f(x) = sin⁻¹(x).
6. f(x) ± f(y) = \(f\left( xy\mp \sqrt{1-{{x}^{2}}}\sqrt{1-{{y}^{2}}} \right)\) is satisfied by f(x) = cos⁻¹(x).
7. f(x) ± f(y) = \(f\left( \frac{x\pm y}{1\mp xy} \right)\) is satisfied by f(x) = tan⁻¹(x).
8. f(x). f(1/x) = f(x) + f(1/x) is satisfied by polynomial function f(x) = ± xⁿ + 1.
Example: let a function f(x) satisfies f(x) + f(2x) + f(2 – x) + f(x + 1) = x ” x ϵ R. then find the value of f(0)
Solution: Given that
f(x) + f(2x) + f (2 – x) + f (x + 1) = x
put x = 0 then
f (0) + f (2 x 0) + f (2 – 0) + f (0 + 1) = 0
f (0) + f (0) + f (2) + f (1) = 0
2f (0) + f (1) + f (2) = 0 … (1)
Put x = 1
f (1) + f (2x 1) + f (2 – 1) + f (1 + 1) = 1
f (1) + f (2) + f (1) + f (2) = 1
2f (1) + 2 f (2) = 1
f (1) + f (2) = ½
from equation (1)
2f (0) + ½ = 0
2 f (0) = -1/2
f (0) = -1/4.