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Functional Equation Solver

Evaluate and solve linear and quadratic functional equations

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Evaluate f(x) at x=k Solve f(x) = c for x

f(x)=ax+b. Find f(k).

f(x)=x+, f()=?

Functional Equations

Evaluate: f(x)=ax+b -> f(k)=a*k+b

Solve f(x)=c: ax+b=c -> x=(c-b)/a

Inverse: f^{-1}(x)=(x-b)/a

Verify: f(f^{-1}(x)) = x

Functional equations relate a function to its values at different points. The simplest is the linear functional equation f(x)=ax+b. Evaluating f(k) is direct substitution. Solving f(x)=c finds the input that gives a specific output.

⚠ The inverse function f^{-1}(x) reverses the mapping: f(f^{-1}(x)) = x. Check by composing both functions.

What Are Functional Equations?

A functional equation defines a function implicitly through relationships between its values. Linear functional equations are the simplest: f(x)=ax+b. Solving them involves substitution, finding inverses, and verifying the function satisfies given conditions.

Evaluation

Given f(x)=ax+b, compute f(k) by substituting k: f(k)=a*k+b. Direct calculation.

Solving

Given f(k)=c and f(x)=ax+b, solve: a*k+b=c -> k=(c-b)/a. Finds the input.

Inverse

Swap x and y, solve for y: f^{-1}(x)=(x-b)/a. Undoes the original function.

Cauchy Equation

f(x+y)=f(x)+f(y) -> f(x)=cx for continuous f. The fundamental linear functional equation.

Teaching Example: f(x)=2x+3, find f(5). f(5)=2*5+3=10+3=13. Solve f(x)=11: 2x+3=11, 2x=8, x=4. Verify: f(4)=2*4+3=11. Correct!

Applications

Algebra Function Theory Composition Inverse Calculus

Frequently Asked Questions

What is functional equation?▼

Equation where unknown is a function. f(x)=ax+b. Evaluate: f(k)=a*k+b. Solve: x=(c-b)/a.

How to evaluate f(k)?▼

Substitute x=k into f(x). For f(x)=ax+b: f(k)=a*k+b. Simple arithmetic.

Solve f(x)=c?▼

Set ax+b=c. Solve for x: x=(c-b)/a. This is the inverse function evaluated at c.

Cauchy equation?▼

f(x+y)=f(x)+f(y). Solution: f(x)=cx for continuous functions. Foundation of linear functions.

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