Session 03-01 - Function Concepts & Business Modeling

Section 03: Functions as Business Models

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Welcome to Functions!

Introduction

  • From equations to functions: A powerful generalization
    • We’ve solved equations; now we’ll model relationships
  • Functions as business tools
    • Cost, revenue, and profit modeling
  • Mathematical used to model real-world applications
    • Essential for business decision-making

Entry Quiz - 10 Minutes

Quick Review from Section 02

Work individually, then we compare together

  1. \[\begin{cases} 2x + 3y = 18 \\ x - y = 1 \end{cases}\]

  2. A company’s costs increase exponentially according to \(C = 1000 \cdot 1.05^t\). After how many years will costs double?

  3. Solve for \(x\): \(\log_2(x+3) + \log_2(x-1) = 3\)

  4. Translate: “The profit equals revenue minus costs, where revenue is 50 euros per unit and costs include a fixed cost of 1000 euros plus 30 euros per unit.”

Homework Discussion - 15 Minutes

Solution from Tasks 02-03

15 minutes - presentation and discussion

  • Present difficult problems
  • Discuss your potential problems
  • Share strategies for your solutions

Your open questions

Let’s discuss Mock Exam 02 and the past sections!

  • Which problem from Mock Exam 02 did you find hardest?
  • Is there something you are not feeling comfortable with?
  • Is there any topic you would like to have repeated?

This is your chance to have something repeated!

Learning Objectives

Learning Objectives

By the end of this session, you can:

  • Explain what makes a relation a function and apply the vertical line test
  • Use function notation to evaluate cost, revenue, and profit functions
  • Determine domain and range - both mathematical and business restrictions
  • Translate between verbal, algebraic, numerical, and graphical representations

From Equations to Functions

What is a Function?

A function is a rule that assigns to each input exactly one output

  • Equation perspective: \(y = 2x + 5\) (a relationship)
    • We’ve been solving these for specific values
  • Function perspective: \(f(x) = 2x + 5\) (a machine)
    • Input any \(x\), get exactly one output \(f(x)\)
  • Business perspective: A model of cause and effect
    • Input: production quantity → Output: total cost

Question: Does this make sense for you?

Function Notation

Symbolic language for modeling

\(f(x) = 2x^2 - 3x + 1\)

  • \(f\) is the function name (like naming a business model)
  • \(x\) is the input variable (independent variable)
  • \(f(x)\) is the output value (dependent variable)

Read as: “f of x equals…”

To find \(f(3)\): \(f(3) = 2(3)^2 - 3(3) + 1 = 18 - 9 + 1 = 10\)

Multiple Function Names

Different functions model different business aspects

  • Cost function: \(C(x)\) = total cost for \(x\) units
  • Revenue function: \(R(x)\) = total revenue from \(x\) units
  • Profit function: \(P(x) = R(x) - C(x)\)
  • Demand function: \(D(p)\) = quantity demanded at price \(p\)

So far not too difficult, right?

Business Example

A bakery has:

  • Fixed costs: 500€ per day
  • Variable costs: 2€ per pastry
  • Selling price: 5€ per pastry

Question: What is the cost, the revenue and the profit function?

Calculator: Defining f(x) and g(x)

Register a function for repeated use:

  1. Press MENU → 9 (Tabellen)
  2. Enter your function expression for \(f(x)\)
  3. You can also define a second function \(g(x)\)
  4. Press = to continue to table settings

Once defined in Tabellen mode, the function is stored for evaluation.

Calculator: Evaluating Functions

Once f(x) is defined in Tabellen mode:

The calculator generates a table showing function values:

  • Set Start, Ende, and Inkre to define x-values
  • The table displays both x and f(x) values

Example: \(f(x) = x^2 - 4x + 3\)

  • Try to compute f(x) from x=0 to x=3 with step 1
  • What can you see?

This helps verify zeros and critical points!

Quick Practice - 10 Minutes

Function Evaluation Practice

Work individually for 5 minutes, then we discuss

Given the functions:

  • \(f(x) = 3x^2 - 2x + 1\)
  • \(g(x) = \frac{x+4}{x-2}\)

Calculate:

  1. \(f(0)\), \(f(2)\), and \(f(-3)\)
  2. \(g(5)\) and \(g(0)\)

Break - 10 Minutes

Domain and Range

Understanding Domain

The domain is the set of all possible input values

  • Mathematical restrictions:
    • Cannot divide by zero
    • Cannot take square root of negative (in real numbers)
    • Logarithm requires positive argument
  • Business restrictions:
    • Cannot produce negative quantities
    • Limited production capacity
    • Budget constraints

Domain Examples

For \(f(x) = \frac{1}{x-3}\):

  • Denominator cannot be zero
  • \(x - 3 \neq 0\)
  • \(x \neq 3\)
  • Domain: \(\mathbb{R} \setminus \{3\}\) or \((-\infty, 3) \cup (3, \infty)\)

For \(g(x) = \sqrt{2x + 6}\):

  • Argument must be non-negative
  • \(2x + 6 \geq 0\)
  • \(x \geq -3\)
  • Domain: \([-3, \infty)\)

Production function \(P(x) = 100\sqrt{x}\) where \(x\) is hours of labor:

  • Mathematical: \(x \geq 0\) (square root)
  • Practical: \(0 \leq x \leq 24\) (hours per day)
  • Domain: \([0, 24]\)

Understanding Range

The range is the set of all possible output values

  • What values can the function actually produce?
  • Often harder to find than domain
  • Depends on the function’s behavior
  • Critical for understanding business limitations

Range = Output!

Finding Range Examples

For \(f(x) = x^2 + 2\):

  • When \(x = 0\): \(f(0) = 2\)
  • Values grow larger as \(|x|\) increases
  • Range: \([2, \infty)\)
  • Note, we’ll learn to find exact minimum points soon!

For \(g(x) = \frac{1}{x}\) with domain \(x \neq 0\):

  • Can be any value except 0
  • Range: \(\mathbb{R} \setminus \{0\}\)

Monthly membership revenue \(R(x) = 50x\) where \(x\) is number of members:

  • Minimum: \(R(0) = 0\) (no members)
  • Increases linearly without bound
  • Range: \([0, \infty)\)
  • Practical limit depends on capacity

Seeing Domain and Range

For \(g(x) = \sqrt{2x + 6}\): domain = shadow on the \(x\)-axis, range = shadow on the \(y\)-axis.

Function Representations

Four Ways to Represent Functions

Each representation offers unique insights

  1. Verbal: “Base costs of 100 which increase by 3 for each additional unit”
  2. Algebraic: \(C(x) = 100 + 3x\)
  3. Numerical: Table of values
  4. Graphical: Visual representation

Question: How would you represent this function as table and as graph?

Example: Mobile Phone Plan

Scenario: Mobile plan costs 20€ base fee plus 0.10€ per minute.

Question: How would you represent this as a function, as a table and as a graph?

Graphical Representation

\(C(m) = 20 + 0.10m\)

The Vertical Line Test

A graph only represents a function if every possible vertical line intersects it at most once

  • Why? Each input must have exactly one output
  • Pass: Linear, quadratic, exponential graphs
  • Fail: Circles, sideways parabolas
  • Business implication: No ambiguity in predictions

Is this a function?

\(y = x^2\)

What about this?

\(x^2 + y^2 = 25\)

The Test at a Glance

One vertical line, one intersection - that is the whole test.

Guided Practice - 25 Minutes

Individual Exercise Block

Work alone for 15 minutes

  1. Find the domain of: \(f(x) = \frac{x+2}{x^2-4}\)

  2. Cost function is \(C(x) = 1500 + 25x\) and revenue is \(R(x) = 40x\)

    1. Find the profit function \(P(x)\)
    2. What is the break-even point?
    3. What is a reasonable domain for this model?

  3. Given the function \(g(x) = 2x + 8\):

    1. Find the domain and range
    2. Find where \(g(x) = 20\)

FSP-Style Practice

10 minutes - exam-style precision

Given \(f(x) = x^2 - 3x\) and the average-cost function \(\bar{C}(x) = \frac{2000 + 25x}{x}\):

  1. Compute and simplify \(f(a + 2)\)

  2. Compute and simplify \(f(2x) - 2f(x)\)

  3. State the domain of \(\bar{C}(x)\) - mathematically and in the business context

Business Applications

Cost Functions in Detail

Understanding the structure of business costs

\[C(x) = \text{Fixed Costs} + \text{Variable Costs}\]

  • Fixed Costs (FC): Rent, insurance, salaries
    • Independent of production level
  • Variable Costs (VC): Materials, hourly wages, utilities
    • Proportional to production
  • Total Cost: \(C(x) = FC + VC \cdot x\)
  • Average Cost: \(AC(x) = \frac{C(x)}{x}\)

Revenue and Demand

The relationship between price and quantity

  • Simple model: \(R(x) = p \cdot x\) (fixed price)
    • Linear revenue function
  • Reality: Price often depends on quantity sold
    • Higher supply → lower price
    • This creates more complex models
  • Preview: In Session 03-03, we’ll explore quadratic models
    • These allow us to find optimal quantities
    • Essential for maximizing profit

Think-Pair-Share - 7 Minutes

Which of these are functions?

Discuss with your neighbour - input, output, and whether each input has exactly one output:

  • Price of a product \(\rightarrow\) quantity demanded
  • Customer \(\rightarrow\) orders placed this month
  • Number of units produced \(\rightarrow\) total cost
  • Revenue \(\rightarrow\) number of units sold

The last one is subtle: two different quantities can generate the same revenue - so reading “units” from “revenue” may not be unique!

Combining Functions

Example: Concert Tickets

  • Fixed price: 40€ per ticket
  • Fixed costs: 10,000€
  • Variable costs: 10€ per ticket
  • Revenue: \(R(x) = 40x\) (linear function)
  • Cost: \(C(x) = 10000 + 10x\) (linear function)
  • Profit: \(P(x) = R(x) - C(x) = 40x - (10000 + 10x)\)
  • Simplified: \(P(x) = 30x - 10000\) (also linear!)
  • Break-even: When \(P(x) = 0\), so \(x = 333.33\) → need 334 tickets

Collaborative Problem-Solving

Group Activity: Startup Analysis

The Scenario

A startup produces custom phone cases:

  • Fixed monthly costs: 3,000€ (rent, equipment, insurance)
  • Material cost per case: 8€
  • Labor cost per case: 7€
  • They plan to sell at a fixed price of 35€ per case

Your Tasks

Work in groups of if you like

  1. Develop the cost function \(C(x)\)
  2. Develop the revenue function \(R(x)\)
  3. Find the profit function \(P(x)\)
  4. Determine the break-even quantity (where \(P(x) = 0\))
  5. If they can produce maximum 500 cases per month, what’s their maximum possible profit?

Coffee Break - 15 Minutes

Practice Tasks

Identifying Functions (x)

Work individually and then we compare.

Determine whether each relation represents a function. If it is not a function, explain why using the vertical line test concept.

  1. \(y = 4x - 7\)
  2. \(x^2 + y^2 = 25\)
  3. \(y = |x - 2|\)
  4. \(x = y^2 + 1\)

Spot the Error

Each statement below contains one mistake - find it!

  1. “The table \(\{(2, 5), (3, 7), (2, 9)\}\) is a function, because every pair has an output.”

  2. “For \(f(x) = \frac{x+2}{x^2-4}\), I cancel \((x+2)\) first, so the domain is \(\mathbb{R} \setminus \{2\}\).”

  3. “If \(f(x) = x^2\), then \(f(a + b) = f(a) + f(b) = a^2 + b^2\).”

Fitness Center Membership Model

A center has collected data on how membership varies with price.

Monthly Fee (€) Number of Members
30 400
40 350
50 300
60 250

The fitness center has:

  • Fixed monthly costs: €15,000
  • Variable costs: €5 per member

Your Tasks

Work in groups for 20 minutes

  1. Show that this data represents a function with monthly fee as input and number of members as output
  2. Write the function \(N(p)\) where \(p\) is the monthly fee (assuming a linear relationship)
  3. Find a reasonable domain for this business context
  4. Write the profit function \(P(p)\) in terms of the monthly fee \(p\)

Wrap-Up & Key Takeaways

Key Takeaways

  • Functions model relationships between variables
  • Domain and range define boundaries of our models
  • Business applications require multiple functions working together
  • The same relationship can be represented in multiple ways
  • Real-world constraints affect mathematical models

Why This Matters for the FSP

Functions are the backbone of the exam

  • Problem 1 (Cost & Profit): builds directly on \(C(x)\), \(R(x)\), \(P(x)\) from today
  • Problem 2 (Function Analysis): starts with domain, notation, and evaluation
  • Every later topic - derivatives, integrals - operates on functions
  • Precise notation and domain thinking earn points on every problem

Final Assessment

5 minutes - Individual work

A local gym has fixed costs of 5000€ per month and variable costs of 10€ per member. They charge 40€ per member per month.

  1. Write the cost function \(C(x)\) where \(x\) is the number of members
  2. Write the revenue function \(R(x)\)
  3. How many members are needed to break even?

Next Session Preview

03-02: Linear Functions & Economic Applications

  • Deep dive into linear functions
  • Supply and demand curves
  • Market equilibrium
  • Linear regression basics
  • Cost-volume-profit analysis

Homework Assignment: Complete Tasks 03-01!