Functional Equations

Plug in the right values — x = 0, y = -x, x 1/x — and the function confesses.

The idea

A functional equation specifies an unknown function by an identity that must hold for every value of its inputs, rather than by a formula. The method for solving one is substitution: because the identity holds for every input, each particular choice of inputs yields a true equation about specific values of the function. Reach for substitution whenever a problem states an identity in an unknown function and asks for one of its values, or for the function itself.

Choose substitutions that make part of the equation collapse into something already known. Setting every input to $0$ (or to $1$, when the equation multiplies rather than adds) usually determines one value to build on. Setting one input to the negative of the other brings in the value just found. Setting the inputs equal, repeatedly, reaches the integer multiples, and dividing works back down to the fractions.

Two cautions apply. A substitution is legal only for inputs the identity was stated for, so respect exclusions such as $x \neq 0$. And substitution determines the function only at inputs the substitutions reach: knowing a function at every rational number does not by itself decide it at every real number — that step needs an extra hypothesis such as continuity.

Ways to work on it

• Walkthrough. Cauchy's equation f(x+y) = f(x) + f(y): substitutions pin down f on the rationals.
• Practice. Substitution drills on additive and multiplicative functional equations.
• Hardest. Solve a functional equation that couples f(x) with f(1/x).

Not sure where to start? Take the ten-question placement test.