Injective Functions

Understanding injectivity is critical for solving functional equations. When a function is injective, we can "cancel" it from both sides of an equation: $f (a) = f (b) ⟹ a = b$ .

Definition

Definition (Injective / One-to-One)

A function

f : A \to B

is injective if different inputs always produce different outputs.

\forall x_{1}, x_{2} \in A, f (x_{1}) = f (x_{2}) ⟹ x_{1} = x_{2}

Horizontal Line Test

Graphically, a function is injective if no horizontal line intersects the graph more than once.

Proving Injectivity

In functional equations, we often prove injectivity to simplify the problem.

Example

Prove that if

f (f (x)) = x + 1

, then

f

is injective.

Proof. Suppose $f (a) = f (b)$ . Apply $f$ to both sides: $f (f (a)) = f (f (b))$ . Substitute the original equation: $a + 1 = b + 1$ . Therefore, $a = b$ . Thus, $f$ is injective. ∎

Strict Monotonicity implies Injectivity

f

is strictly increasing (

x < y ⟹ f (x) < f (y)

) or strictly decreasing, it is automatically injective.

Practice Problems

Exercise (Problem 1)

Determine if

f (x) = x^{3}

is injective.

Exercise (Problem 2)

Determine if

f (x) = x^{2}

is injective on the domain

R

. What if the domain is

[0, \infty)

Exercise (Problem 3)

Let

f : R \to R

satisfy

f (x + f (y)) = f (x) + y

. Prove that

f

is injective.