**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) = a^{x}
as f (x + y) = a^{x+y} = a^{x} a^{y} = f(x) f(y).

2. f (x + y) = f (x – y) = f(x)/f(y) is satisfied by f (x) =
a^{x} as f (x -y) = a^{x-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) = log_{a}x
+ log_{a}b

log_{a}xy = f (xy)

4. f(x) – f(y) = f(x/y) is satisfied by f(x) = log_{e}x
as f(x) – f(y) = log_{e}x – log_{e}y \(={{\log }_{e}}\left( \frac{x}{y} \right)=f\left(
\frac{x}{y} \right)\),

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.