Functional Equations Resulting from Properties of Functions
1. Odd functions having symmetry of graph about the origin: f(x) + f(-x) = 0, \( \forall \) x ϵ Df.
2. Even functions having symmetry of graph about the y – axis: f(x) = f(-x), \( \forall \) x ϵ Df.
3. Symmetry of graph about the Point (a, 0): f(a – x) = – f(a + x).
4. Symmetry of graph about the line x = a: f(a – x) = f(a + x).
5. Periodic functions: f(x) = f(x + T) \( \forall \) x ϵ Df for least positive value T.
Example: If f: R→R is an odd function such that
(i) f(1+x) = 1+ f(x)
(ii) x² f(1/x) = f(x), x ≠ 0, Then find f(x).
Solution: Given that f(1 + x) = 1 + f(x)
x² f(1/x) = f(x), x ≠ 0
Replace x by x + 1, then
\({{(1+x)}^{2}}f\left( \frac{1}{1+x} \right)=f(x+1)\),
Given condition f(1+x) = 1+ f(x)
\(f\left( \frac{1}{1+x} \right)=\frac{1+f(x)}{{{(1+x)}^{2}}}\) … (1)
\(f\left( \frac{1}{1+x} \right)=f\left( 1-\frac{x}{1+x} \right)\),
\(=1+f\left( -\frac{x}{1+x} \right)\),
\(=1-f\left( \frac{x}{1+x} \right)\) (∵ f(x) is odd)
\(=1-{{\left( \frac{x}{1+x} \right)}^{2}}f\left( \frac{1+x}{x} \right)\),
\(=1-{{\left( \frac{x}{1+x} \right)}^{2}}f\left( 1+\frac{1}{x} \right)\) … (2)
From (1) and (2) equations
\(\frac{1+f(x)}{{{(1+x)}^{2}}}=1-{{\left( \frac{x}{1+x} \right)}^{2}}\left( 1+\frac{f(x)}{{{x}^{2}}} \right)\),
1 + f (x) = (1 + x)² – x² – f (x)
f(x) = x.