Properties of Functions

Why This Matters

Functions are the backbone of abstract mathematics—they formalize the idea of a relationship between sets and give us precise language to describe how mathematical objects connect. In this course, you're being tested on your ability to classify functions by their structural properties, understand when inverses exist, and work with function composition as a fundamental operation. These concepts appear everywhere: in proofs about set cardinality, in group theory, and in virtually every branch of higher mathematics.

Don't just memorize definitions—know what each property tells you about a function's behavior. When you see "injective," you should immediately think "distinct inputs, distinct outputs—so inverses are possible on the range." When you see "surjective," think "the codomain is fully covered." Master these conceptual connections, and you'll handle both definition-recall questions and proof-based problems with confidence.

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Defining a Function's Scope

Before analyzing what a function does, you need to understand where it operates. These foundational concepts establish the sets a function connects and distinguish between what's theoretically possible and what actually occurs.

Domain and Codomain

• Domain is the set of all valid inputs—the "source" set from which every element must map somewhere
• Codomain is the declared set of potential outputs, which may include elements never actually hit by the function
• Distinguishing these matters because two functions with identical rules but different codomains are technically different functions

Range (Image)

• Range is the actual set of outputs produced—the subset of the codomain that gets "hit"
• Always satisfies $\text{Range}(f) \subseteq \text{Codomain}(f)$, with equality only when $f$ is surjective
• Computing the range often requires solving $f(x) = y$ and determining which $y$-values have solutions

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Injectivity, Surjectivity, and Bijectivity

These three properties are the classification system for functions. They tell you about the mapping behavior between domain and codomain and determine whether inverses exist. Mastering these distinctions is non-negotiable for proofs involving functions.

One-to-One (Injective) Functions

• Injective means distinct inputs always produce distinct outputs: $f(a) = f(b) \Rightarrow a = b$
• No collisions occur—each codomain element is hit at most once by the function
• Proof strategy: to show injectivity, assume $f(a) = f(b)$ and derive $a = b$

Onto (Surjective) Functions

• Surjective means every codomain element is hit at least once: $\forall y \in \text{Codomain}, \exists x \in \text{Domain}: f(x) = y$
• Range equals codomain—nothing in the target set is "missed"
• Proof strategy: for arbitrary $y$ in codomain, explicitly construct an $x$ with $f(x) = y$

Bijective Functions

• Bijective means both injective and surjective—a perfect one-to-one correspondence
• Each codomain element is hit exactly once, creating a complete pairing between sets
• Guarantees invertibility—only bijective functions have true two-sided inverses

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Inverses and Composition

These operations let you reverse functions and chain them together. Understanding when inverses exist and how composition behaves is essential for working with function equations and proving properties about combined functions.

Inverse Functions

• Inverse $f^{-1}$ satisfies $f^{-1}(f(x)) = x$ and $f(f^{-1}(y)) = y$—it "undoes" the original function
• Existence requires bijectivity—without injectivity, $f^{-1}$ wouldn't be well-defined; without surjectivity, $f^{-1}$ wouldn't have full domain
• Notation warning: $f^{-1}(x)$ means the inverse function, not $\frac{1}{f(x)}$

Composition of Functions

• Composition $(f \circ g)(x) = f(g(x))$ applies $g$ first, then $f$ to the result
• Domain requirement: the range of $g$ must be contained in the domain of $f$
• Not commutative—generally $f \circ g \neq g \circ f$, which matters for proofs

Identity Function

• Identity $\text{id}(x) = x$ maps every element to itself—the "do nothing" function
• Neutral element for composition: $f \circ \text{id} = \text{id} \circ f = f$
• Inverse relationship: $f \circ f^{-1} = f^{-1} \circ f = \text{id}$ defines what "inverse" means

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Special Function Types

Certain functions have distinctive behaviors that simplify analysis or appear frequently in applications. Recognizing these types helps you quickly identify properties without detailed computation.

Constant Functions

• Constant function $f(x) = c$ outputs the same value regardless of input
• Never injective (unless domain has one element)—multiple inputs map to identical output
• Surjective only if codomain equals $\{c\}$; otherwise fails surjectivity

Monotonic Functions

• Increasing: $x_1 < x_2 \Rightarrow f(x_1) \leq f(x_2)$; strictly increasing uses $<$ instead of $\leq$
• Decreasing: $x_1 < x_2 \Rightarrow f(x_1) \geq f(x_2)$; strictly decreasing uses $>$
• Strictly monotonic functions are injective—this is a key proof technique for establishing one-to-one behavior

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Symmetry and Periodicity

These properties describe structural patterns in how functions behave. They're especially important for simplifying computations and understanding function graphs.

Even and Odd Functions

• Even functions satisfy $f(-x) = f(x)$—symmetric about the y-axis
• Odd functions satisfy $f(-x) = -f(x)$—symmetric about the origin
• Useful fact: the only function that's both even and odd is $f(x) = 0$

Periodic Functions

• Periodic with period $T$ means $f(x + T) = f(x)$ for all $x$ in the domain
• Smallest positive $T$ with this property is called the fundamental period
• Never injective on unbounded domains—values repeat, so distinct inputs share outputs

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Continuity and Discontinuity

These properties describe whether a function's graph has "gaps." While rigorous treatment requires limits, the intuition is essential for understanding function behavior.

Continuous Functions

• Continuous at $a$ means $\lim_{x \to a} f(x) = f(a)$—no jumps, breaks, or holes
• Continuous on a set means continuous at every point in that set
• Composition preserves continuity—if $f$ and $g$ are continuous, so is $f \circ g$

Discontinuous Functions

• Removable discontinuity: limit exists but doesn't equal function value (or function undefined)
• Jump discontinuity: left and right limits exist but differ
• Essential (infinite) discontinuity: at least one side has no limit or approaches infinity

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Function Notation

Clear notation is the foundation of precise mathematical communication. Sloppy notation leads to errors and lost points.

Function Notation and Evaluation

• Standard notation $f: A \to B$ declares domain $A$, codomain $B$, and function name $f$
• Evaluation $f(a)$ means substituting $a$ into the function rule to compute the output
• Arrow notation $x \mapsto x^2$ describes the rule without naming the function

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Quick Reference Table

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Self-Check Questions

1. A function $f: \mathbb{R} \to \mathbb{R}$ is strictly increasing. Without any other information, what can you conclude about its injectivity and surjectivity? Explain why.

2. Compare and contrast injective and surjective functions: what does each guarantee about how domain elements map to codomain elements? Give an example of a function that is one but not the other.

3. Why must a function be bijective (not just injective) to have a true inverse function? What goes wrong if we try to define an inverse for a non-surjective function?

4. If $f$ and $g$ are both bijective, prove or explain why $f \circ g$ must also be bijective. Which property of composition makes this work?

5. A function is both even and periodic with period $T$. What additional symmetry does this create? Can such a function ever be injective on $\mathbb{R}$?