Image and Codomain

Context: FIT1058_MOC · the two “output sets” of a function · image $\subseteq$ codomain · the gap is closed by surjectivity

Quick Revision

• 🎯 Objective: codomain = declared output set; image = values actually produced ➔ image $\subseteq$ codomain.
• 📦 Core Components: codomain (promise) ➔ image $\{f(x)\}$ (truth).
• ⚡ Key Constraint: image = codomain iff surjective; the exact image is often intractable.

📝 Core

1. Two Output Sets

• Codomain ➔ declared output space $B$, guaranteed to contain every $f(x)$.
• Image ➔ $\text{Image}(f)=\{f(x):x\in A\}$, the exact set of values.
• Inclusion ➔ image $\subseteq$ codomain (possibly proper).

2. Promise vs Truth

• Codomain promises ➔ all outputs lie in $B$, but not that every element of $B$ is hit.
• Why generous ➔ the exact image can be hard/impossible to pin down (sum-of-four-cubes is open).
• Examples ➔ Fibonacci image $\{1,2,3,5,8,\dots \}⊊\mathbb{N}$; $x^2$ image ${\mathbb{R}}_0^{+}⊊\mathbb{R}$.

3. When They Coincide

• Surjective ➔ image = codomain ➔ every codomain element is used.
• Terminology ➔ avoid “range” (ambiguous); use codomain / image.

Key identities:

$\text{Image}(f)=\{f(x):x\in A\}\subseteq B=\text{codomain}$ $\text{surjective}⟺\text{Image}(f)=\text{codomain}$

⚖️ Core Decision Matrix

Set	Meaning	Determined by
codomain	declared output space	the specification
image	achieved values	the rule
relation	image $\subseteq$ codomain	always
equality	image = codomain	surjective

When It Flips: the codomain is chosen for convenience; the image is forced by the rule. "Surjective" is exactly "the codomain promise is met with nothing unused". The preimages $f^{-1}(y)$ partition the domain (Equivalence Relation).

📊 Exam Execution Trace

Manual Execution Trace

$f:\{-1,0,1,2\}\to \mathbb{Z}$, $f(x)=x^2$:

Step / State	$x$	$f(x)$	Image so far
0 (Init)	—	—	$\varnothing$
1	$-1,1$	1	$\{1\}$
2	0	0	$\{0,1\}$
3	2	4	$\{0,1,4\}$

⚠️ Common Mistakes

• 💡 Codomain must contain all values ➔ a set smaller than the image is not a valid codomain; a generous codomain (e.g. $\mathbb{Z}$) is chosen because the exact image may be intractable.

🧠 Active Recall

Distinguish codomain and image, and why do we usually specify a codomain rather than the image?

• Hint: Promise vs truth.

Answer

• Short answer: Codomain = declared output set (promise); image = exact values produced.
• Why: Intractable image ➔ a clean superset like $\mathbb{Z}$ is easy to state and still valid (sum-of-four-cubes is open).

Why avoid "range", and when does image equal codomain?

• Hint: Surjectivity.

Answer

• Short answer: “Range” is ambiguous (image or codomain); image = codomain exactly when surjective.
• Why: Onto ➔ every codomain element is some $f(x)$, no unused elements.