Proofs & Properties of Functions: Injectivity, Surjectivity, Bijectivity, and Functional Equations

Core Concepts & Definitions

1. Types of Functions

A function $f:A\to B$ maps each element of set $A$ (domain) to exactly one element in set $B$ (codomain). Three key classifications:

classDiagram
    class Function {
        <<abstract>>
        +f: A → B
    }
    class Injective {
        +One-to-one
        +f(a) = f(b) ⇒ a = b
    }
    class Surjective {
        +Onto
        +∀y ∈ B, ∃x ∈ A: f(x) = y
    }
    class Bijective {
        +Injective + Surjective
        +Inverse exists
    }
    Function <|-- Injective
    Function <|-- Surjective
    Injective <|-- Bijective
    Surjective <|-- Bijective

Visual Comparison:

2. Proving Injectivity/Surjectivity/Bijectivity

Method 1: Definition-Based Proof

• Injective: Assume $f(a)=f(b)$, show $a=b$.
• Surjective: For any $y\in B$, find $x\in A$ such that $f(x)=y$.
• Bijective: Prove both injective and surjective.

Method 2: Horizontal/Vertical Line Test

• Injective: No two $x$-values map to the same $y$ (passes horizontal line test).
• Surjective: Every $y$-value in codomain is covered (passes vertical line test if codomain is $\mathbb{R}$).

Worked Examples with Real-World Ties

Example 1: Proving $f(x)=3x+5$ is Bijective

Proof:

1. Injective: Assume $f(a)=f(b)$: $3a+5=3b+5\Rightarrow 3a=3b\Rightarrow a=b$. Visual: The graph is a straight line with slope 3 (strictly increasing).

2. Surjective: For any $y\in \mathbb{R}$, solve $y=3x+5$: $x=\frac{y-5}{3}$. Since $x\in \mathbb{R}$, $f$ covers all $y$.

Real-World Tie:

• eSewa’s Transaction ID Mapping: Each transaction ID (input) maps uniquely to a payment status (output). If $f$ is bijective, no two IDs share the same status, and every status corresponds to exactly one ID.

  flowchart TD
    A["Transaction ID (x)"] -->|"f(x)"| B["Payment Status (y)"]
    B -->|"f⁻¹(y)"| A

Example 2: Domain and Range of $f(x)=\sqrt{6-x-x^2}$

Step 1: Find Domain The expression under the square root must be non-negative: $6-x-x^2\ge 0$. Rewrite: $-x^2-x+6\ge 0$ or $x^2+x-6\le 0$. Factor: $(x+3)(x-2)\le 0$. Solution: $x\in [-3,2]$.

Step 2: Find Range Let $y=\sqrt{6-x-x^2}$. The maximum of $6-x-x^2$ occurs at $x=-b/(2a)=0.5$: $y_{\text{max}}=\sqrt{6-0.5-(0.5{)}^2}=\sqrt{5.25}\approx 2.29$. At endpoints: $y(-3)=0$, $y(2)=0$. Range: $[0,\sqrt{5.25}]$.

Real-World Tie:

• Pathao’s Delivery Time Estimation: The function models the time $y$ (range) a rider takes to deliver an order based on distance $x$ (domain). The domain $[-3,2]$ could represent a bounded delivery zone (e.g., 3 km radius), while the range shows possible delivery times (0 to ~2.3 hours).

Functional Equations

Example 3: Solving $f\left(\frac{2ab}{1+a^2{b}^2}\right)=2f(ab)$

Given $f(x)=\log \frac{1-x}{1+x}$, verify the equation for $|ab|<1$.

Step 1: Substitute $x=ab$ Let $u=ab$. Then: $$f\left(\frac{2u}{1+u^2}\right)=\log \frac{1-\frac{2u}{1+u^2}}{1+\frac{2u}{1+u^2}}=\log \frac{(1-u{)}^2}{(1+u{)}^2}=2\log \frac{1-u}{1+u}=2f(u).$$

Graph of $f(x)$:

Real-World Tie:

• Khalti’s Transaction Logs: Suppose $f(x)$ represents the log of a transaction’s success probability. The functional equation could model how combining two transactions (via $ab$) affects the combined success log. For example, if $a$ and $b$ are two payment attempts, $\frac{2ab}{1+a^2{b}^2}$ might represent a weighted success metric.

Irrational Numbers and Proof Techniques

Example 4: Prove $\sqrt{2}$ is Irrational

Proof by Contradiction:

1. Assume $\sqrt{2}=\frac{p}{q}$ where $p,q$ are coprime integers.
2. Then $2=\frac{p^2}{q^2}\Rightarrow p^2=2q^2$.
3. $p^2$ is even ⇒ $p$ is even ⇒ $p=2k$.
4. Substitute: $(2k{)}^2=2q^2\Rightarrow 4k^2=2q^2\Rightarrow q^2=2k^2$.
5. $q^2$ is even ⇒ $q$ is even. Contradicts coprimality.

Visual:

Real-World Tie:

• NEPSE’s Share Price Calculations: Many financial models rely on irrational numbers (e.g., $\sqrt{2}$ in portfolio optimization). Proving irrationality ensures certain algorithms (like those for calculating optimal trade sizes) avoid rounding errors that could arise with rational approximations.

Domain and Range: Additional Examples

Example 5: $f(x)=\frac{2x+1}{3-x}$

Domain: Denominator $\ne 0\Rightarrow x\ne 3$. Range: Let $y=\frac{2x+1}{3-x}$. Solve for $x$: $y(3-x)=2x+1\Rightarrow 3y-yx=2x+1\Rightarrow x=\frac{3y-1}{y+2}$. Since $x\ne 3$, $y\ne 1$. Thus, range is $\mathbb{R}\setminus \{1\}$.

Graph:

Real-World Tie:

• Bank Loan Interest Rates: The function could model the relationship between loan amount $x$ (domain) and monthly payment $y$ (range). The hole at $x=3$ might represent a maximum loan cap, while $y\ne 1$ could indicate a minimum payment threshold.

## In the Real World

1. eSewa’s Bijective Mapping:

   • Idea: Bijectivity ensures each user has a unique transaction record.
   • How: The system uses injective functions to map user IDs to transactions and surjective functions to ensure every transaction is logged (no gaps).

2. Pathao’s Delivery Time Function:

   • Idea: Domain/range analysis models delivery feasibility.
   • How: The domain $[-3,2]$ (km) limits delivery zones, while the range $[0,2.3]$ hours sets rider expectations.

3. Ncell’s Network Coverage:

   • Idea: Functional equations optimize signal distribution.
   • How: Equations like $f(ab)$ might model combined signal strength from two towers, ensuring seamless coverage.

## Exam Tip

1. Proofs:

   • Always start with definitions (e.g., "Assume $f(a)=f(b)$" for injectivity).
   • Use contradiction for irrationality proofs (as in $\sqrt{2}$).

2. Domain/Range:

   • For square roots: Solve $\text{expression}\ge 0$.
   • For rational functions: Exclude values making denominator zero; solve $y=f(x)$ for $x$ to find range.

3. Functional Equations:

   • Substitute specific forms (e.g., $x=ab$) to verify identities.
   • Graph the function to visualize behavior (e.g., $f(x)=\log \frac{1-x}{1+x}$).

4. Common Pitfalls:

   • Forgetting to check codomain for surjectivity.
   • Misapplying the horizontal/vertical line test (e.g., confusing injective/surjective graphs).
   • In domain problems, ignoring square root/denominator constraints.

## Practice Questions (TU-Style)

1. Prove $f(x)=5x-2$ is bijective.
2. Find the domain and range of $f(x)=\sqrt{9-x^2}$.
3. Verify $f\left(\frac{x}{1+x^2}\right)=2f(x)$ for $f(x)=\arctan x$.
4. Show $\sqrt{3}$ is irrational using proof by contradiction.
5. For $f(x)=\frac{x^2-1}{x-2}$, find domain and range.

Answer Key:

1. Injective: $5a-2=5b-2\Rightarrow a=b$. Surjective: $x=(y+2)/5$.
2. Domain: $[-3,3]$; Range: $[0,3]$.
3. Substitute $x=\frac{u}{1+u^2}$, simplify using $\arctan$ addition formula.
4. Assume $\sqrt{3}=p/q$, derive $p^2=3q^2$, show both $p$ and $q$ even.
5. Domain: $x\ne 2$; Range: $\mathbb{R}\setminus \{3\}$.