Course Cheatsheet

Section 03: Functions as Business Models

What is a Function?

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

  • Mathematical Definition: For every input \(x\), there is exactly one output \(f(x)\)
  • Business Perspective: A model showing cause and effect relationships
  • Key Principle: Each input must have exactly one output (no ambiguity)

Function Notation

Traditional Equation Function Notation Meaning
\(y = 2x + 5\) \(f(x) = 2x + 5\) Function \(f\) maps input \(x\) to output \(2x + 5\)
\(y = x^2 - 3\) \(g(x) = x^2 - 3\) Function \(g\) maps input \(x\) to output \(x^2 - 3\)
\(y = \frac{100}{x}\) \(h(x) = \frac{100}{x}\) Function \(h\) maps input \(x\) to output \(\frac{100}{x}\)

Function Evaluation:

  • \(f(3)\) means “substitute \(x = 3\) into function \(f\)”
  • If \(f(x) = 2x + 5\), then \(f(3) = 2(3) + 5 = 11\)
  • If \(g(x) = x^2 - 3\), then \(g(-2) = (-2)^2 - 3 = 1\)

Domain and Range

Domain: All Possible Input Values

Mathematical Restrictions:

Function Type Restriction Domain Rule
Rational: \(f(x) = \frac{1}{x-a}\) Denominator ≠ 0 \(x \neq a\)
Square Root: \(f(x) = \sqrt{x-a}\) Argument ≥ 0 \(x \geq a\)
Logarithm: \(f(x) = \log(x-a)\) Argument > 0 \(x > a\)
Polynomial: \(f(x) = ax^n + ...\) No restrictions All real numbers

Business Restrictions:

  • Quantities cannot be negative: \(x \geq 0\)
  • Production capacity limits: \(0 \leq x \leq \text{max capacity}\)
  • Time constraints: \(0 \leq t \leq 24\) hours per day

Range: All Possible Output Values

Common Patterns:

Function Type Range Example
Linear: \(f(x) = ax + b\) All real numbers \((-\infty, \infty)\)
Quadratic: \(f(x) = x^2 + c\) \([c, \infty)\) \(f(x) = x^2 + 2\) has range \([2, \infty)\)
Square Root: \(f(x) = \sqrt{x}\) \([0, \infty)\) Non-negative outputs
Rational: \(f(x) = \frac{1}{x}\) All except 0 \((-\infty, 0) \cup (0, \infty)\)

Four Ways to Represent Functions

1. Verbal Description

“Base cost of 100€ plus 3€ for each additional unit”

2. Algebraic Formula

\(C(x) = 100 + 3x\)

3. Numerical Table

\(x\) \(C(x)\)
0 100
10 130
20 160

4. Graphical Plot

Visual representation showing the relationship between input and output

The Vertical Line Test

Rule: A graph represents a function if and only if every vertical line intersects it at most once.

Examples:

  • Pass: Lines, parabolas opening up/down, exponential curves
  • Fail: Circles, sideways parabolas, vertical lines

Why Important: Ensures each input has exactly one output (function definition)

Function Evaluation Strategies

Direct Substitution

If \(f(x) = 3x^2 - 5x + 2\), find \(f(4)\): - \(f(4) = 3(4)^2 - 5(4) + 2 = 3(16) - 20 + 2 = 48 - 20 + 2 = 30\)

Piecewise Functions

When function has different rules for different input ranges:

\[f(x) = \begin{cases} 2x + 1 & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}\]

  • \(f(-3) = 2(-3) + 1 = -5\) (use first rule since \(-3 < 0\))
  • \(f(5) = 5^2 = 25\) (use second rule since \(5 \geq 0\))

Key Vocabulary

  • Function: Rule assigning exactly one output to each input
  • Domain: Set of all possible input values
  • Range: Set of all possible output values
  • Independent Variable: Input variable (usually \(x\))
  • Dependent Variable: Output variable (usually \(y\) or \(f(x)\))
  • Function Notation: \(f(x)\) read as “\(f\) of \(x\)”
  • Vertical Line Test: Method to determine if graph represents function
  • Fixed Costs: Business costs independent of production level
  • Variable Costs: Business costs proportional to production level
  • Break-Even Point: Where revenue equals cost (profit = 0)

Linear Functions in Detail

Forms of Linear Functions

1. Slope-Intercept Form: \(y = mx + b\)

  • m: slope (rate of change/marginal change)
    • Positive: increasing function
    • Negative: decreasing function
    • Zero: horizontal line
  • b: y-intercept (starting value/base value)
    • Value when \(x = 0\)
    • Often represents fixed costs or initial values

2. Point-Slope Form: \(y - y_1 = m(x - x_1)\)

  • Useful when you know one point \((x_1, y_1)\) and the slope \(m\)
  • Best for modeling from observed data

Converting Between Forms:

  • From two points \((x_1, y_1)\) and \((x_2, y_2)\):
    • Slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
    • Use point-slope form, then simplify to slope-intercept

Supply and Demand Functions

Demand Functions

Demand shows how quantity purchased depends on price

  • General form: \(Q_d = a - bp\) (decreasing function)
    • \(a\): maximum quantity when price = 0
    • \(b\): price sensitivity (how much demand drops per unit price increase)
  • Alternative form: \(p = c - dQ_d\) (price as function of quantity)

Supply Functions

Supply shows how quantity produced depends on price

  • General form: \(Q_s = -c + dp\) (increasing function)
    • Often has positive y-intercept
    • \(d\): production response to price changes

Market Equilibrium

Equilibrium occurs where supply equals demand: \(Q_d = Q_s\)

  • Equilibrium price (\(p^*\)): Market-clearing price
  • Equilibrium quantity (\(Q^*\)): Amount actually traded
  • Graphically: Intersection point of supply and demand curves
  • No shortage or surplus at equilibrium

Example:

  • Demand: \(Q_d = 500 - 50p\)
  • Supply: \(Q_s = -100 + 100p\)
  • Set equal: \(500 - 50p = -100 + 100p\)
  • Solve: \(600 = 150p\), so \(p^* = 4\)
  • Quantity: \(Q^* = 500 - 50(4) = 300\) units

Cost-Volume-Profit

CVP Framework Components

Component Formula Description
Fixed Costs (FC) Constant Independent of production volume
Variable Costs per unit (VC) Per unit amount Changes with production
Total Costs \(TC = FC + VC \times Q\) Sum of fixed and variable costs
Revenue \(R = P \times Q\) Price × Quantity
Profit \(\Pi = R - TC\) Revenue minus total costs
Contribution Margin \(CM = P - VC\) Profit per unit before fixed costs

CVP Key Calculations

Break-Even Point: \(Q_{BE} = \frac{FC}{CM} = \frac{FC}{P - VC}\)

Target Profit: \(Q_{target} = \frac{FC + \text{Target Profit}}{CM}\)

Margin of Safety: Actual sales - Break-even sales

Linear Modeling from Data

Steps to Create Linear Models

  1. Identify variables (independent vs dependent)
  2. Calculate slope between data points
  3. Use point-slope form to find equation
  4. Interpret parameters in business context

Example: Sales Forecasting

  • Data shows consistent increase of 15 units per month
  • Starting at 120 units in month 1
  • Model: \(S(m) = 15m + 105\)

Depreciation Models

Linear (Straight-Line) Depreciation

Formula: \(V(t) = V_0 - dt\)

Where:

  • \(V(t)\): Value at time \(t\)
  • \(V_0\): Initial value
  • \(d\): Depreciation rate per period
  • Useful life: \(n = \frac{V_0}{d}\) periods

Example: Company Vehicle

  • Purchase price: €30,000
  • Annual depreciation: €5,000
  • Function: \(V(t) = 30,000 - 5,000t\)
  • Fully depreciated after 6 years

Quadratic Functions

Standard Form

Formula: \(f(x) = ax^2 + bx + c\)

Key Components: - a: Direction and width - \(a > 0\): Opens upward (U-shape, has minimum) - \(a < 0\): Opens downward (∩-shape, has maximum) - \(|a|\) larger → Narrower parabola - b: Affects position of vertex - c: y-intercept (value when \(x = 0\))

Graph Shape: Parabola (curved, not straight like linear functions)

The Vertex Formula

Key Formula: \(x_v = -\frac{b}{2a}\)

For quadratic function \(f(x) = ax^2 + bx + c\): - Vertex x-coordinate: \(x_v = -\frac{b}{2a}\) - Vertex y-coordinate: \(f(x_v) = f(-\frac{b}{2a})\) - Vertex represents: - Maximum if \(a < 0\) (parabola opens down) - Minimum if \(a > 0\) (parabola opens up) - Axis of symmetry: Vertical line \(x = x_v\)

Vertex Form

Alternative representation: \(f(x) = a(x - h)^2 + k\)

  • Vertex: \((h, k)\) - directly visible!
  • Direction: \(a\) (same as standard form)
  • Advantage: Vertex immediately apparent
  • Examples:
    • \(f(x) = 2(x - 3)^2 + 5\) → Vertex at \((3, 5)\), minimum
    • \(g(x) = -(x + 4)^2 + 10\) → Vertex at \((-4, 10)\), maximum

Completing the Square

Convert from standard form to vertex form

Process:

  1. Factor out \(a\) from first two terms
  2. Complete the square inside parentheses: add and subtract \((\frac{b}{2a})^2\)
  3. Simplify to vertex form \(a(x - h)^2 + k\)

Example: Convert \(f(x) = 2x^2 - 12x + 10\)

  1. Factor out 2: \(f(x) = 2(x^2 - 6x) + 10\)
  2. Complete square: Need \((\frac{-6}{2})^2 = 9\)
  3. Add/subtract 9: \(f(x) = 2(x^2 - 6x + 9 - 9) + 10\)
  4. Factor perfect square: \(f(x) = 2((x - 3)^2 - 9) + 10\)
  5. Distribute: \(f(x) = 2(x - 3)^2 - 18 + 10\)
  6. Final form: \(f(x) = 2(x - 3)^2 - 8\) → Vertex at \((3, -8)\)

Business Optimization with Quadratic Functions

Revenue Optimization (Price-Dependent Demand)

When demand depends on price: \(Q = a - bp\)

  • Revenue: \(R(p) = p \times Q = p(a - bp) = ap - bp^2\)
  • This creates a quadratic function in price
  • Optimal price: \(p^* = -\frac{a}{2(-b)} = \frac{a}{2b}\)

Profit vs Revenue Maximization

Important Distinction:

  • Revenue maximization: Ignores costs, focuses on \(R(p) = p \times Q(p)\)
  • Profit maximization: Includes costs, \(\Pi(p) = R(p) - C(p)\)
  • Different optimal points: Revenue-maximizing price ≠ Profit-maximizing price

Cost Functions (Quadratic)

Some costs increase quadratically with production:

  • Formula: \(C(x) = ax^2 + bx + c\)
  • Interpretation:
    • Fixed costs: \(c\)
    • Increasing marginal costs: \(a > 0\)
  • Minimum cost production: \(x^* = -\frac{b}{2a}\) (if \(a > 0\))

Profit Functions (Quadratic)

Common form: \(P(x) = -ax^2 + bx + c\) (where \(a > 0\))

  • Opens downward: Has maximum profit
  • Optimal production: \(x^* = \frac{b}{2a}\)
  • Break-even points: Solve \(P(x) = 0\)

Advanced Applications

Area Optimization

Classic problem: Maximize area with constraint

Example: Rectangular storage with 200m fencing - One side against building (no fence needed) - Let \(x\) = width, \(y\) = length along building - Constraint: \(2x + y = 200\) → \(y = 200 - 2x\) - Area: \(A(x) = x \cdot y = x(200 - 2x) = 200x - 2x^2\) - Maximum: \(x^* = \frac{200}{2(2)} = 50m\) - Optimal dimensions: 50m × 100m = 5,000 m²

Projectile Motion in Business

Applications: Product launch campaigns, market penetration

  • Formula: \(h(t) = -\frac{g}{2}t^2 + v_0t + h_0\)
  • Business analog: \(A(t) = -at^2 + bt + c\) (awareness over time)
  • Peak timing: \(t^* = \frac{b}{2a}\)

Example: Marketing Campaign - Awareness: \(A(t) = -2t^2 + 24t\) - Peak at: \(t = \frac{24}{2(2)} = 6\) weeks - Maximum awareness: \(A(6) = 72\) points

Market Analysis Steps

  1. Identify the quadratic relationship (revenue, profit, cost)
  2. Find the vertex using \(x = -\frac{b}{2a}\)
  3. Calculate optimal value by substituting back
  4. Check constraints (production limits, market capacity)
  5. Consider break-even points (solve \(f(x) = 0\))

Key Business Insights

  • Symmetric behavior: Equal results at equal distances from optimum
  • Diminishing returns: Beyond optimal point, additional input reduces output
  • Trade-offs: Revenue maximization vs profit maximization
  • Constraints matter: Mathematical optimum may not be practically achievable

Function Transformations

Vertical Transformations

Moving graphs up or down

Transformation Formula Effect Business Example
Vertical Shift Up \(g(x) = f(x) + k\) (k > 0) Entire graph moves up Fixed cost increase
Vertical Shift Down \(g(x) = f(x) - k\) (k > 0) Entire graph moves down Government subsidy
Vertical Stretch \(g(x) = a \cdot f(x)\) (a > 1) Graph becomes taller Price increase across all products
Vertical Compression \(g(x) = a \cdot f(x)\) (0 < a < 1) Graph becomes shorter Discount pricing
Reflection over x-axis \(g(x) = -f(x)\) Graph flips upside down Revenue becomes loss

Business Applications:

  • Fixed Cost Changes: \(C_{new}(x) = C(x) + k\) (rent increase of k)
  • Percentage Markups: \(R_{new}(x) = 1.2 \cdot R(x)\) (20% price increase)
  • Tax Effects: \(P_{net}(x) = 0.8 \cdot P(x)\) (20% tax rate)

Horizontal Transformations

Moving graphs left or right

Transformation Formula Effect Business Example
Horizontal Shift Right \(g(x) = f(x - h)\) (h > 0) Graph moves right Market entry delay
Horizontal Shift Left \(g(x) = f(x + h)\) (h > 0) Graph moves left Early product launch
Horizontal Stretch \(g(x) = f(x/b)\) (b > 1) Graph becomes wider Extended product lifecycle
Horizontal Compression \(g(x) = f(bx)\) (b > 1) Graph becomes narrower Accelerated lifecycle
Reflection over y-axis \(g(x) = f(-x)\) Graph flips left-right Reverse time analysis

Key Point: Horizontal transformations are counterintuitive!

  • \(f(x - 3)\) shifts RIGHT by 3 units
  • \(f(x + 3)\) shifts LEFT by 3 units
  • \(f(2x)\) compresses (makes narrower)
  • \(f(x/2)\) stretches (makes wider)

Business Applications:

  • Seasonal Adjustments: \(D(t) = f(t - 3)\) (peak shifts 3 months later)
  • Market Speed Changes: \(L(t) = f(2t)\) (2x faster product lifecycle)
  • Time Scaling: Converting quarterly to monthly data

Combining Transformations

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

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

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

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

Reading Economic Graphs

Key Features to Identify

Graph Analysis Checklist:

  • Intercepts: Starting values, break-even points
  • Slope/Rate of change: Marginal values, trends
  • Maximum/Minimum: Optimal points, extremes
  • Intersections: Equilibrium, equal values
  • Shape: Linear, quadratic, exponential patterns
  • Domain/Range: Feasible regions, constraints

Business Graph Interpretation

Cost vs Revenue Analysis:

  • y-intercept of cost: Fixed costs
  • Intersection points: Break-even quantities
  • Region between intersections: Profitable zone
  • Maximum vertical distance: Optimal production level
  • Slope comparisons: Marginal cost vs marginal revenue

Market Analysis:

  • Supply/Demand intersection: Market equilibrium
  • Shift patterns: Market changes over time
  • Area under curves: Total consumer/producer surplus
  • Steep vs flat curves: Price sensitivity (elasticity)

Function Composition

Understanding Composition

Composition models sequential processes: \((f \circ g)(x) = f(g(x))\)

Key Points:

  • Read as “f composed with g”
  • Apply g first, then f to the result
  • Output of g becomes input of f
  • Order matters: \((f \circ g) \neq (g \circ f)\) usually
  • Models multi-step business processes

Steps to Find Composition:

  1. Calculate \(g(x)\) first
  2. Substitute result into \(f\)
  3. Simplify the expression

Common Business Compositions

  • Manufacturing: Raw materials → Components → Finished products
  • Finance: Local currency → Foreign currency → Investment returns
  • Retail: Wholesale → Retail price → After-tax price
  • Marketing: Leads → Conversions → Revenue

Domain Considerations

When composing functions, the domain of \((f \circ g)\) is restricted by:

  • Domain of g (inner function)
  • Values of g(x) must be in domain of f (outer function)

Inverse Functions

What is an Inverse Function?

An inverse function reverses the original function

Definition: If \(f(a) = b\), then \(f^{-1}(b) = a\)

  • The inverse “undoes” what the original function does
  • Notation: \(f^{-1}(x)\) (read as “f inverse of x”)
  • Not the same as \(\frac{1}{f(x)}\) (reciprocal)

Testing for Invertibility

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

Horizontal Line Test:

  • Each horizontal line intersects the graph at most once
  • Equivalently: each output comes from exactly one input
  • For continuous functions: must be always increasing or always decreasing

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

Business Applications of Inverse Functions

Common Business Inverses:

  • Demand ↔︎ Price: \(Q = 1000 - 20p\) → \(p = \frac{1000 - Q}{20}\)
  • Cost ↔︎ Quantity: \(C = 500 + 25x\) → \(x = \frac{C - 500}{25}\)
  • Profit ↔︎ Sales: Find required sales for target profit
  • Break-even Analysis: Reverse engineer production requirements

Graphical Properties of Inverses

  • Reflection property: Graph of \(f^{-1}\) is reflection of \(f\) over line \(y = x\)
  • Domain and range swap: Domain of f = Range of \(f^{-1}\)
  • Intersection points: \(f\) and \(f^{-1}\) intersect on line \(y = x\)

Problem Solving

Systematic Problem-Solving Approach

  1. Identify the context: What business situation is modeled?
  2. Determine function type: Linear, quadratic, composed, inverse?
  3. Identify variables: Input (independent) vs output (dependent)
  4. Check domain restrictions: Mathematical and business constraints
  5. Write the function: Express relationship algebraically with units
  6. Apply appropriate techniques: Vertex formula, composition, transformations
  7. Verify results: Check if outputs make business sense
  8. Consider practical constraints: Production limits, budget restrictions

Common Pitfalls to Avoid

  • Composition order: Remember \((f \circ g) \neq (g \circ f)\)
  • Domain issues: Always check for restrictions
  • Units confusion: Track currency vs. quantity units carefully
  • Inverse vs. reciprocal: \(f^{-1}(x) \neq \frac{1}{f(x)}\)
  • Optimization assumptions: Verify that mathematical optimum is practical