Session 03-04 - Transformations, Composition & Inverses

Section 03: Functions as Business Models

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

Review from Session 03-03

Work individually, then we compare together

1. Find the vertex of \(f(x) = -2x^2 + 12x - 10\) and determine if it’s a maximum or minimum.

2. A company’s profit function is \(P(x) = -x^2 + 80x - 1200\). Find:

   1. The quantity that maximizes profit
   2. The maximum profit

Review from Session 03-03 II

3. Convert \(g(x) = x^2 - 4x + 7\) to vertex form.

4. Revenue is modeled by \(R(p) = p(600 - 3p)\). What price maximizes revenue?

Homework Discussion

Sharing Solutions from Tasks 03-03

Focus on optimization strategies

• Problem 7: Theatre ticket prices
  • How did the 600 seats change the Saturday price?
• Problem 9: Fireworks
  • Vertex form from the highest point?
• FSP Problem: Wallboxes
  • Why is the profit maximum below the revenue maximum?

Key Insight

Optimization often involves balancing mathematical ideals with practical constraints!

Learning Objectives

Learning Objectives

By the end of this session, you can:

• Apply vertical and horizontal shifts, stretches, and reflections to business functions
• Combine multiple transformations in the correct order
• Compose functions to model multi-step business processes
• Find and interpret inverse functions for prices, costs, and demand

Vertical Changes

Vertical Shifts

Moving graphs up or down

Given original function \(f(x)\):

• Upward shift: \(g(x) = f(x) + k\) (k > 0)
• Downward shift: \(g(x) = f(x) - k\) (k > 0)
• Graph effect: Entire graph moves vertically
• Business meaning:
  • Fixed cost changes
  • Base price adjustments
  • Overhead modifications

Example: Cost Function Adjustment

Original cost: \(C(x) = 5x^2 + 3x + 100\)

• Rent increases by 100 EUR: \(C_{new}(x) = 5x^2 + 3x + 200\)
• Government subsidy of 50 EUR: \(C_{new}(x) = 5x^2 + 3x + 50\)

Question: Any idea how we could graph this?

Vertical Shift Visualization

Same parabola shape, shifted vertically - fixed costs change, but variable cost structure remains the same!

Vertical Stretch & Compression

Changing the vertical scale

Given original function \(f(x)\):

• Vertical stretch: \(g(x) = a \cdot f(x)\) where \(a > 1\)
• Vertical compression: \(g(x) = a \cdot f(x)\) where \(0 < a < 1\)
• Reflection: \(g(x) = -f(x)\) (flip over x-axis)
• Business meaning:
  • Percentage markups/discounts
  • Tax multipliers
  • Currency conversions

Example: Revenue Scaling

Original revenue: \(R(x) = 50x - 0.5x^2\)

• 20% price increase across all products: \(R_{new}(x) = 1.2(50x - 0.5x^2) = 60x - 0.6x^2\)
• 20% price decrease across all products: \(R_{new}(x) = 0.8(50x - 0.5x^2) = 40x - 0.4x^2\)

Question: Can anyone describe what happens now?

All three functions have the same optimal quantity (50 units), but revenue scales proportionally with price - stretch up or compress down!

Vertical Stretch and Compression

Vertical Stretch & Compression: Price Changes

Reflection Visualization

Reflection: Flipping Over x-axis

Quick Practice

Work individually, then we discuss

Given the original profit function: \(P(x) = -x^2 + 40x - 200\)

1. Write the new function if fixed costs increase by 100 EUR
2. Write the new function if a government grant reduces costs by 50 EUR
3. Write the new function if all profits increase by 30%
4. Write the new function if all profits decrease by 25%
5. What business scenario could \(-P(x)\) represent?
6. Write the reflected function explicitly

Horizontal Changes

Horizontal Shifts

Moving graphs left or right

Given original function \(f(x)\):

• Right shift: \(g(x) = f(x - h)\) where \(h > 0\)
• Left shift: \(g(x) = f(x + h)\) where \(h > 0\)
• Business meaning:
  • Time delays or advances
  • Market entry timing, seasonal adjustments

Counterintuitive: Minus shifts right, plus shifts left!

Example: Seasonal Demand Shift

Summer demand peaks in June (month 6): \(D(t) = -(t-6)^2 + 100\)

• Unusually hot weather shifts peak to May (left shift): \(D_{new}(t) = -(t-5)^2 + 100\)
• Unusually cold weather shifts peak to July (right shift): \(D_{new}(t) = -(t-7)^2 + 100\)

Question: Anyone with an idea how to graph this?

Same shape parabola, shifted horizontally - peak demand moves but pattern stays the same!

Horizontal Shift Visualization

Horizontal Shifts: Seasonal Demand Changes

Remember: \(-(t-5)^2 + 100\) peaks at \(t = 5\) (May, shifted LEFT), \(-(t-7)^2 + 100\) peaks at \(t = 7\) (July, shifted RIGHT) - the number after the minus sign is the peak, not the direction!

Horizontal Stretch & Compression

Changing the horizontal scale

Given original function \(f(x)\):

• Horizontal stretch: \(g(x) = f(x/b)\) where \(b > 1\)
• Horizontal compression: \(g(x) = f(bx)\) where \(b > 1\)
• Reflection: \(g(x) = f(-x)\) (flip over y-axis)
• Business meaning:
  • Time scaling (quarterly to monthly)
  • Production speed changes
  • Market cycle adjustments

Example: Product Lifecycle

Original lifecycle (monthly): \(L(t) = -t^2 + 8t + 1000\)

• Competitor speeds up cycle (2x faster): \(L_{fast}(t) = -(2t)^2 + 8(2t) + 1000 = -4t^2 + 16t + 1000\)
• Extended warranty slows cycle (2x slower): \(L_{slow}(t) = -(t/2)^2 + 8(t/2) + 1000 = -0.25t^2 + 4t + 1000\)

Question: What happens to the lifecycle duration?

Horizontal compression (faster) → narrower curve, earlier peak.

Horizontal stretch (slower) → wider curve, later peak!

Counterintuitive again: \(f(2t)\) compresses (faster), \(f(t/2)\) stretches (slower)!

Horizontal Stretch/Compression

Horizontal Scaling: Product Lifecycle Changes

Combined Changes

Order of Operations

Apply transformations systematically

Standard order for \(g(x) = a \cdot f(b(x - h)) + k\):

1. Horizontal stretch/compress by factor \(b\)
2. Horizontal shift by \(h\)
3. Vertical stretch/compress by factor \(a\)
4. Vertical shift by \(k\)

Let’s apply these steps to a function!

Complete Transformation Example

Start with \(f(x) = x^2\)

Transform to: \(g(x) = -2(x - 3)^2 + 5\)

• Shift right 3 units: \((x - 3)^2\)
• Stretch vertically by 2: \(2(x - 3)^2\)
• Reflect over x-axis: \(-2(x - 3)^2\)
• Shift up 5 units: \(-2(x - 3)^2 + 5\)

Question: Who can describe how this might look like?

Original Function

Starting Point: \(f(x) = x^2\)

Progressive Transformation

All Transformation Steps

Scenario: Market Expansion

Original profit in Germany: \(P(x) = -x^2 + 40x - 180\)

Research estimates expansion to France with adjustments:

• 20% lower profit: Multiply by 0.8
• Fixed cost increase of 200 EUR: Subtract 200
• 3-month delay: Replace \(x\) with \((x - 3)\)
• \(P_{France}(x) = 0.8[-(x-3)^2 + 40(x-3) - 180] - 200\)

Question: Should the company expand to France?

Market Expansion Visualization

Germany vs France: Profit Comparison

Guided Practice

Individual Exercise Block I

Work alone for 10 minutes, then we discuss the solutions

1. Given \(f(x) = x^2 - 4x + 3\), write the equation for:

   1. \(f(x)\) shifted up 5 units
   2. \(f(x)\) shifted left 2 units
   3. \(f(x)\) reflected over the x-axis

2. A cost function is \(C(x) = 0.5x^2 + 20x + 500\). Due to inflation:

   • All costs increase by 10%
   • An additional fixed cost of 100 EUR is added
   • Write new cost function and find cost for producing 50 units.

Individual Exercise Block II

Work alone for 5 minutes, then we discuss the solutions

1. The demand for ice cream follows \(D(t) = -2(t - 7)^2 + 200\) where \(t\) is the month.
   1. In which month is demand highest?
   2. If climate change shifts the peak 1 month earlier and increases maximum demand by 15%, write the new function.

Break

Function Composition

Understanding Composition

Composition models sequential processes

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

• Read as “f composed with g”
• Apply g first, then f
• Output of g becomes input of f
• Order matters! \((f \circ g) \neq (g \circ f)\) usually
• Business meaning: Multi-step processes

Composition: Supply Chain

Imagine a company manufacturing products from raw materials.

1. Raw materials to components: \(g(x) = 2x + 10\) (cost in EUR)
2. Components to products: \(f(y) = 3y + 50\) (cost in EUR)

• \((f \circ g)(x) = f(g(x)) = f(2x + 10)\)
• \(= 3(2x + 10) + 50\)
• \(= 6x + 30 + 50\)
• \(= 6x + 80\)
• For 10 units raw material: Cost = 140 EUR

Supply Chain Visualization

Function Composition: Supply Chain Cost Analysis

Domain Considerations

Domain of composition can be restricted

For \((f \circ g)(x)\):

1. Start with domain of \(g\)
2. Find range of \(g\)
3. Intersect with domain of \(f\)
4. Track back to valid \(x\) values

This is not too complicated, right? Here we just need to be careful.

Example: Dosage & Safe Maximum

Calculate dosage based on body weight, then check safety limit

• Dosage from weight: \(g(x) = 5x\) mg (5mg per kg body weight)
• Safe processing: \(f(x) = \sqrt{500 - x}\) (requires total dose \(\leq 500\) mg)
• Composition requires: \(5x \leq 500\), so \(x \leq 100\) kg body weight

Therefore, we cannot process a patient weighing more than 100kg, as this would be unsafe!

Quick Practice

Work individually, then we discuss

A food delivery service has the following cost structure:

• Restaurant to warehouse: \(g(x) = 1.5x + 5\) (EUR per order, where \(x\) is number of orders)
• Warehouse to customer: \(f(y) = 2y + 8\) (EUR delivery cost)

1. Find the composition \((f \circ g)(x)\) representing total delivery cost.

2. Calculate the total cost for 20 orders.

3. If the domain of \(g\) is \([0; 100]\) orders, what is the range of \(g\)?

Inverse Functions

What is an Inverse Function?

An inverse function reverses the original function

If \(f(a) = b\), then \(f^{-1}(b) = a\). The inverse “undoes” what the original function does.

Testing for Invertibility

A function has an inverse if it’s one-to-one

• Each output comes from exactly one input
• Horizontal line test: Each horizontal line hits graph at most once
• For continuous functions: Always increasing or always decreasing

Question: Which of these two fulfill the one-to-one condition?

• \(f(x) = 2x + 5\)
• \(g(x) = x^2\)

Just make sure, you don’t use the vertical line test here!

Finding Inverse Functions

Step-by-step process

1. Replace \(f(x)\) with \(y\)
2. Swap \(x\) and \(y\)
3. Solve for \(y\)
4. Replace \(y\) with \(f^{-1}(x)\)
5. Verify domain and range

Looks complicated? It is actually rather easy!

Example: Simple Linear Function

Find the inverse of \(f(x) = 3x + 6\)

1. \(y = 3x + 6\)
2. \(x = 3y + 6\) (swap \(x\) and \(y\))
3. \(3y = x - 6\) (solve for \(y\))
4. \(y = \frac{x - 6}{3}\)
5. Inverse: \(f^{-1}(x) = \frac{x - 6}{3}\)

Verify: \(f(f^{-1}(x)) = f\left(\frac{x-6}{3}\right) = 3 \cdot \frac{x-6}{3} + 6 = x - 6 + 6 = x\) ✓

Visualizing the Inverse

Function \(f(x) = 3x + 6\) and its inverse \(f^{-1}(x) = (x-6)/3\) reflected over \(y=x\)

Example: Price-Demand Inverse

Demand function: \(Q = 1000 - 20p\)

1. \(y = 1000 - 20p\)
2. \(x = 1000 - 20p\) (swap)
3. \(x - 1000 = -20p\)
4. \(p = 50 - 0.05x\)
5. Inverse: \(p(Q) = 50 - 0.05Q\)

What does this mean? It gives the price needed for specific quantity!

Visualizing Price and Demand

Demand function and its inverse reflected over the line \(y=x\).

Guided Practice

Individual Exercise Block

Work alone for 15 minutes, then discuss for 10 minutes

1. Given \(f(x) = 2x + 3\) and \(g(x) = x^2 - 1\):

   1. Find \((f \circ g)(x)\) and evaluate \((f \circ g)(2)\)
   2. Find \((g \circ f)(x)\) and evaluate \((g \circ f)(2)\)

2. A company converts raw materials through two stages:

   • Stage 1: \(C_1(x) = 50x + 200\) (process cost in EUR)
   • Stage 2: \(C_2(y) = 2y + 500\) (assembly cost in EUR)
   • Find the total cost function and cost of 100 units of material.

3. Find the inverse of \(f(x) = \frac{2x + 3}{5}\) and verify your answer.

Team Problem-Solving

International Business Model I

The Scenario: Global E-Commerce Platform

An e-commerce company operates internationally with these functions:

Pricing Model:

• Base price in USD: \(P(x) = 50 + 0.8x\) where \(x\) is quantity
• EUR conversion: \(E(p) = 0.85p\)
• UK markup: \(U(p) = 1.2p + 10\)

International Business Model II

The Scenario: Global E-Commerce Platform

Shipping Costs:

• Weight calculation: \(W(x) = 0.5x + 2\) (kg)
• Shipping cost: \(S(w) = 15w + 25\) (EUR)

Customer Demand:

• At total price \(T\): \(D(T) = 5000 - 20T\) units per month

Your Tasks

If you like, you can work in groups

1. Find the total price function for EU customers (product + shipping) as a function of quantity \(x\).

2. Find the inverse of the demand function. What does it represent?

3. Create a composite function that gives monthly demand based on quantity ordered.

4. If the company wants exactly 2000 units demanded per month, what should the quantity per order be?

5. The company can only process orders where total price leads to positive demand. Find the maximum viable order quantity.

Coffee Break

FSP Workshop

Problem 1: A Bath Bomb Press

A small manufactory presses bath bombs (fizzing balls for the bathtub) and sells them to drugstores. For each bath bomb, the drugstores pay 4 EUR more than its ingredients and packaging cost. When the press runs \(h\) hours per week, its weekly running costs (staff, energy, maintenance, overtime) are \(K(h) = 20h^2 + 640h + 8000\) EUR, with \(0 \leq h \leq 60\). The press makes 400 bath bombs per hour. Let \(x\) be the weekly output in QU (1 QU = 100 bath bombs).

1. Give the running time \(h(x)\) in hours. Determine the running costs \(L(x) = K(h(x))\) as a function of the output, with their domain. Describe how the graph of \(L\) arises from the graph of \(K\). [4 pts.]

Bath Bombs: Maximum Profit

2. Give the weekly profit \(P(x)\) in EUR. Compute the maximum profit and prove that it really is a maximum. Give the output in bath bombs and the running time of the press. [4 pts.]

(for verification: \(L(x) = 1.25x^2 + 160x + 8000\))

Bath Bombs: A Faster Press

A new mould makes the press 25 % faster: it now makes 500 bath bombs per hour. The running costs \(K(h)\) for \(h\) hours stay the same.

3. Show that the new running costs are \(L_2(x) = L\left(\frac{x}{1.25}\right)\). Describe this transformation and give \(L_2(x)\) in standard form. [3 pts.]

4. The production manager expects: “The press is 25 % faster, so the best output also rises by 25 %, to 12,000 bath bombs per week.” Determine the new profit-maximizing output, the running time and the maximum profit. Assess the manager’s expectation. [5 pts.]

Wrap-Up & Takeaways

Key Takeaways

• Vertical shifts represent fixed changes, horizontal shifts timing changes
• Stretches and compressions show scaling effects
• Composition models sequential processes - order matters: \((f \circ g) \neq (g \circ f)\)
• Inverses reverse calculations; one-to-one functions are invertible
• Multiple transformations and function chains model complex business scenarios

Final Assessment

5 minutes - Individual work

A retailer has:

• Cost function: \(C(x) = 20x + 500\)
• Price function: \(P(x) = 50 - 0.5x\)
• Revenue: \(R(x) = x \cdot P(x)\)

1. Express revenue as a function of \(x\) explicitly
2. Find the inverse of the cost function
3. If the retailer has a budget of 2,500 EUR, how many units can they stock?

Next Session: Mock Exam 04

Session 03-05 is exam only - no lecture

• 90 minutes working time plus 5 minutes reading time, 50 points
• Covers all of Section 03: functions, linear and quadratic models, transformations, composition, inverses
• Allowed: non-programmable calculator, drawing instruments, monolingual dictionary
• Not allowed: notes, formula sheets, any electronic devices
• Counts toward the 60% of test points needed for exam admission

Preparing for Mock Exam 04

Homework: complete Tasks 03-04, then prepare for Mock Exam 04 with FSP Training 03 (sidebar, Section 03 worksheets).