Week-05

Maths 1: Week 05 - The Master Encyclopedia of Function Mappings

1. The Genesis: The Machine Metaphor 📜

1.1 The Evolution of Interaction

In Weeks 1 through 4, we looked at functions as standalone entities—Lines, Parabolas, and Polynomials. But in the real world, systems interact. The output of one process becomes the input of the next. Think of a computer program where a user’s click (Input A) triggers a calculation (Process 1), which then triggers a visual update (Process 2).

This is the power of Function Mapping. We stop looking at just the formula and start looking at the Flow of Logic. This week, we explore the “Advanced Grammar” of functions: how they combine (Composition), how they mirror (Symmetry), and how they can be reversed (Inverses).

1.2 The Philosophical Intuition

If a Function is a “Machine,” then Composition is a factory assembly line, and the Inverse is an “Undo” button. This week is about mastering the controls of these machines.

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2. Axiomatic Foundations: The Logic of Mapping 🏛️

2.1 The Domain & Range Revisited

A function $f:A\to B$ is a rule that assigns every $x\in A$ to exactly one $f(x)\in B$.

1. Domain ($A$): The set of all legal inputs.
2. Codomain ($B$): The set of all potential outputs.
3. Range: The set of all actual outputs. Axiom: For a mapping to be a function, every input must have one, and only one, output.

2.2 The “Existence” Triad

• Injective (1-to-1): Every output has at most one unique input. (No two $x$ values share a $y$).
• Surjective (Onto): The Range equals the Codomain. (Every $y$ value is reached by at least one $x$).
• Bijective: Both Injective and Surjective. This is a “Perfect Marriage” of sets.

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3. The Topology of Combination: Composition 🛠️

3.1 Composite Functions ($f\circ g$)

Definition: $(f\circ g)(x)=f(g(x))$.

• Visual: You drop $x$ into the $g$ machine, and then drop the result into the $f$ machine.

3.2 The Strict Domain Requirement

The domain of $f\circ g$ is NOT just the domain of $x$. It is a set of $x$ such that:

1. $x$ is in the Domain of $g$.
2. $g(x)$ is in the Domain of $f$. Intuition: A pass-code is only valid if it works at BOTH security checkpoints.

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4. The Mirror of Logic: Symmetry 🖋️

4.1 Even vs. Odd Functions

Functions possess structural balance that can simplify calculus and physics.

• Even Functions: Perfectly balanced like a human face (Mirror Symmetry across the Y-axis).
  • Rule: $f(-x)=f(x)$.
• Odd Functions: Perfectly balanced like a propeller (Rotational Symmetry about the Origin).
  • Rule: $f(-x)=-f(x)$.

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5. The Inverse: The “Undo” Operation 📈

5.1 The Condition of Invertibility

A function has an inverse if and only if it is Bijective.

• The Horizontal Line Test: If any horizontal line hits the graph more than once, the function is NOT invertible.
• Why? Because if one $y$ comes from two different $x$s, the inverse wouldn’t know which $x$ to go back to.

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6. The Encyclopedia of Worked Examples (10 Case Studies) 📚

Case 1: Finding Domain of a Composite Function

Problem: $f(x)=\frac{1}{x-1},g(x)=\sqrt{x}$. Find the domain of $(f\circ g)(x)$.

• Step 1: $x$ must be $\ge 0$ (for $g(x)$ to work).
• Step 2: $g(x)$ must not be 1 (because $f(1)$ is undefined). So $\sqrt{x}\ne 1⟹x\ne 1$.
• Result: Domain is $[0,1)\cup (1,\infty )$.

Case 2: Proving Symmetry for Polynomials

Problem: Is $f(x)=3x^4-2x^2+5$ even, odd, or neither?

• Logic: All powers are even. $f(-x)=3(-x{)}^4-2(-x{)}^2+5=3x^4-2x^2+5$.
• Result: Even.

Case 3: Algebraic Derivative of an Inverse

Problem: If $f(x)=2x+1$, find $f^{-1}(x)$.

• Procedure:
  • $y=2x+1$
  • Swap: $x=2y+1$
  • Isolate $y$: $y=(x-1)/2$.
• Result: $f^{-1}(x)=0.5x-0.5$.

Case 4: The Injective Test

Problem: Is $f(x)=|x|$ injective on $\mathbb{R}$?

• Logic: $f(1)=1$ and $f(-1)=1$. Two inputs give the same output.
• Result: No.

Case 5: Vertical and Horizontal Shifting

Problem: What happens to the graph of $y=f(x)$ if we change it to $y=f(x-2)+5$?

• Logic: $x-2$ shifts it Right 2. $+5$ shifts it Up 5.

Case 6: Domain of Logarithms

Problem: Find the domain of $f(x)=\ln (x-5)$.

• Logic: The argument of a log must be $>0$.
• Result: $x>5$.

Case 7: Composite Swap ($f\circ g$ vs $g\circ f$)

Problem: $f(x)=x^2,g(x)=x+3$. Are $f(g(x))$ and $g(f(x))$ same?

• Calculation:
  • $f(g(x))=(x+3{)}^2=x^2+6x+9$.
  • $g(f(x))=x^2+3$.
• Result: No. Composition order matters!

Case 8: Proving Surjectivity

Problem: Is $f(x)=x^3$ surjective from $\mathbb{R}\to \mathbb{R}$?

• Logic: For every real number $y$, does an $x$ exist such that $x^3=y$? Yes, $x=\sqrt[3]{y}$.
• Result: Yes.

Case 9: Symmetry of $1/x$

Problem: Is $f(x)=1/x$ even or odd?

• Logic: $f(-x)=1/(-x)=-(1/x)$.
• Result: Odd. (Look at the graph—it rotates perfectly around the origin).

Case 10: Inverse of Square Root

Problem: Find the inverse of $f(x)=\sqrt{x-2}$.

• Logic: Swap: $x=\sqrt{y-2}$. Square both sides: $x^2=y-2⟹y=x^2+2$.
• Constraint: Must keep the original domain in mind. The range of the original was $y\ge 0$, so the domain of the inverse is $x\ge 0$.
• Result: $f^{-1}(x)=x^2+2$ for $x\ge 0$.

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7. Fundamental “How-To” Recipes 🍳

Recipe: Identifying the Parent Function

1. Strip away shifts: Remove all $+5,-2,\dots$
2. Strip away scales: Remove the $2x,3x,\dots$
3. Identify the Core: You are left with $x,x^2,x^3,\sqrt{x},1/x,$ etc. Knowing the parent graph allows you to sketch any modification instantly.

Recipe: Proving Invertibility (The Full Check)

1. Algebraic Check: Solve $f(x_1)=f(x_2)$. If you get $x_1=x_2$ exclusively, it’s injective.
2. Calculus Check: If the derivative $f^{\prime }(x)$ is always positive (or always negative), the graph only ever goes up (or down). It will never hit the same height twice!

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8. Encyclopedia Mastery Challenge 🏆

1. The Double Inverse: What is $(f^{-1}{)}^{-1}(x)$? Explain why this makes logical sense.
2. The Odd Multiplier: If you multiply two Odd functions, is the result Even or Odd? (Hint: Think about the signs).
3. The Hidden Range: If $f(x)=\sin (x)+5$, what is its range? Does it have an inverse on the domain $[-\pi ,\pi ]$?
4. The Self-Inverse: Find a function $f(x)$ such that $f(x)=f^{-1}(x)$ and $f(x)\ne x$. (Hint: Think about 1/x).

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🚀 Master Status: You have completed the Encyclopedic Expansion of Function Mappings. You now understand the flow of logical machines and how to reverse them at will. 助