Functions, Composition & Inverses: Types, Properties & Applications

TAKEAWAYS:

• Injective vs. Surjective vs. Bijective: Master the definitions and visual tests (horizontal/vertical line tests) to classify functions.
• Composition ≠ Multiplication: Understand (f∘g)(x) means "apply g first, then f"—trace with tables to avoid mistakes.
• Inverses Exist Only for Bijective Functions: Learn how to algebraically derive f⁻¹(x) and verify it using f(f⁻¹(x)) = x.
• Real-World Mappings: Functions model database joins (SQL), encryption (Khalti), and pathfinding (Pathao routes)—connect theory to apps.
• Proof Techniques: Use contradiction and direct substitution to prove injectivity/surjectivity.
• Exam Pitfalls: Avoid assuming (f∘g)⁻¹ = g⁻¹∘f⁻¹ without proof; always verify domains/codomains.

1. Types of Functions: Injective, Surjective, Bijective

Definitions with Visual Tests

A function f: A → B can be classified by how it maps elements of A to B:

Worked Example: Classify and Prove

Problem: Show f: ℝ → ℝ, f(x) = 5x − 7 is bijective. Find f⁻¹(x).

Solution:

1. Injective Proof (Contradiction): Assume f(a) = f(b). Then: $$5a-7=5b-7⟹5a=5b⟹a=b.$$ Thus, injective.

2. Surjective Proof (Existence): For any y ∈ ℝ, solve y = 5x − 7: $$x=\frac{y+7}{5}\in R.$$ Thus, surjective.

3. Inverse Function: Swap x and y in y = 5x − 7: $$x=5y-7⟹y=\frac{x+7}{5}.$$ So, f⁻¹(x) = (x + 7)/5.

Verification:

Key Insight: Inverses undo the original function.

2. Function Composition: (f∘g)(x)

How It Works

Composition f∘g means "apply g first, then f": $$(f\circ g)(x)=f(g(x)).$$

Caution: Order matters! (f∘g) ≠ (g∘f) unless f and g commute.

Worked Example: Real-World Scenario

Problem: Pathao’s ride-hailing system uses two functions:

• g(x): Estimates fare based on distance x (km): g(x) = 10x + 50.
• f(y): Adds a 10% surge fee: f(y) = 1.1y. Find (f∘g)(x) and interpret it.

Solution:

1. Compute (f∘g)(x): $$f(g(x))=1.1(10x+50)=11x+55.$$
2. Interpretation:
   • g(x): Base fare (e.g., 5 km → ₹100).
   • f(g(x)): Total fare with surge (₹110).

Visual Trace:

flowchart TD
    A["Distance (x km)"] -->|"g(x)"| B["Base Fare (₹10x+50)"]
    B -->|"f(y)"| C["Total Fare (₹1.1y)"]
    C --> D["(f∘g)(x) = 11x + 55"]

3. Inverse Functions and Their Properties

When Does an Inverse Exist?

Only bijective functions have inverses. For non-bijective functions:

• Injective but not surjective: Restrict codomain to make it bijective (e.g., f(x) = x² on [0, ∞)).
• Surjective but not injective: No inverse unless you allow relations (multivalued functions).

Worked Example: Bank Loan Interest

Problem: A bank offers a loan with interest calculated by f(P) = P(1 + 0.05t), where P is principal, t is time (years). Find f⁻¹ to determine principal from total repayment.

Solution:

1. Rewrite y = P(1 + 0.05t) as P = y / (1 + 0.05t). But to find f⁻¹, treat t as fixed: $$f^{-1}(y)=\frac{y}{1.05}\text{(for fixed }t=1\text{ year)}.$$
2. Verification: $$f(f^{-1}(y))=\frac{y}{1.05}\times 1.05=y.$$

Real-World Tie:

• Khalti’s Transaction ID: Acts like an injective function mapping user actions to unique IDs.
• NEPSE Stock Codes: Bijective mapping between company names and ticker symbols (e.g., NICASIA → NICASIA).

4. Composition of Inverses

Key Theorem

If f and g are bijective: $$(f\circ g{)}^{-1}=g^{-1}\circ f^{-1}.$$

Proof:

graph LR
    A["(f∘g)⁻¹"] -->|"Definition"| B["x such that (f∘g)(x) = y"]
    B --> C["g(x) = f⁻¹(y)"]
    C --> D["x = g⁻¹(f⁻¹(y))"]
    D --> E["Thus, (f∘g)⁻¹(y) = g⁻¹(f⁻¹(y))"]

Worked Example: Encryption (Khalti-Style)

Problem: Let f(x) = 3x + 2 (encrypt) and g(x) = x/2 (decrypt). Show (f∘g)⁻¹ = g⁻¹∘f⁻¹.

Solution:

1. Compute f⁻¹ and g⁻¹:
   • f⁻¹(x) = (x − 2)/3.
   • g⁻¹(x) = 2x.
2. Compute (f∘g)⁻¹ directly: $$(f\circ g)(x)=f(g(x))=3(x/2)+2=(3x/2)+2.$$ Inverse: x = (3y/2) + 2 ⇒ y = (2x − 4)/3.
3. Compute g⁻¹∘f⁻¹: $$g^{-1}(f^{-1}(x))=2\times \frac{x-2}{3}=\frac{2x-4}{3}.$$ Matches!

5. Common Mistakes and Pitfalls

1. Assuming (f∘g)⁻¹ = f⁻¹∘g⁻¹:
   • Wrong: Order matters! Correct: (f∘g)⁻¹ = g⁻¹∘f⁻¹.
2. Ignoring Domains/Codomains:
   • Example: f: [0,∞) → ℝ, f(x) = x² is bijective only if codomain is restricted to [0,∞).
3. Composition vs. Multiplication:
   • (f∘g)(x) ≠ f(x) * g(x). Always evaluate step-by-step.

In the Real World

1. Pathao’s Ride Matching:

   • Function: f(user_id) → driver_id (injective, since one user gets one driver).
   • Composition: g(driver_id) → location ∘ f(user_id) maps a user to their driver’s location.
   • Inverse: f⁻¹(driver_id) → user_id (used for dispute resolution).

2. Khalti’s Transaction IDs:

   • Bijective Function: f(user_action) → unique_transaction_ID.
   • Why Bijective? Ensures no two actions share an ID (injective) and every action gets an ID (surjective).
   • Inverse: f⁻¹(ID) → user_action (used for refunds).

3. Daraz’s Order Queue:

   • Partial Order: Orders are processed based on time + priority (not a total order).
   • Function Composition: g(order) → delivery_status ∘ f(order) → warehouse_assignment tracks order flow.

Exam Tip

1. For Proofs:

   • Injective: Use contradiction (f(a) = f(b) ⇒ a = b).
   • Surjective: Show for any y ∈ B, there exists x ∈ A such that f(x) = y.
   • Bijective: Combine both proofs or use the inverse.

2. For Composition:

   • Always substitute step-by-step. Example: Given f(x) = x² + 1, g(x) = 2x: $$(f\circ g)(x)=f(g(x))=f(2x)=(2x{)}^2+1=4x^2+1.$$
   • Never write (f∘g)(x) = f(x) * g(x).

3. Inverse Functions:

   • To find f⁻¹(x), solve y = f(x) for x in terms of y.
   • Verify: Check f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

4. Diagrams Save Marks:

   • Draw mapping diagrams for injective/surjective functions.
   • Use flowcharts for composition (e.g., Pathao’s fare calculation).

5. Real-World Questions:

   • Expect problems like:
     • "A bank charges interest as f(P) = P(1 + r)². If r = 0.05, find the inverse to calculate principal from maturity amount."
     • "Pathao’s surge pricing is f(x) = 1.2x. If base fare is g(d) = 10d + 50, express total fare as a composition."

Final Reminder:

"A function is like a vending machine: injective means no two inputs give the same output; surjective means every snack is available; bijective means every input gives a unique snack, and you can reverse the process." — Discrete Math Proverb