6️⃣ Functions Useful in Economics and Management — Slides

Mathematical Fundamentals · MS01-001-G

Learning Objectives

  • Determine the domain of definition of a function and identify forbidden values
  • Describe and graph the usual functions (affine, polynomial, logarithm, exponential)
  • Solve simple and general second-degree equations
  • Choose the right function type to model an economic phenomenon

1. Functions — Definition and Domain

Definition

A function \(f: x \mapsto f(x)\) associates each input \(x\) in the domain \(D_f\) to a unique output \(f(x)\).

Forbidden values are values of \(x\) that make \(f(x)\) undefined — e.g. division by zero, or the square root of a negative number.

Example: \(f(x) = \dfrac{1}{x-3}\)

  • Domain: \(\mathbb{R} \setminus \{3\}\)
  • \(x = 3\) is a forbidden value — it makes the denominator zero

2. Usual Functions

Function Formula Shape Economic use
Affine \(f(x) = ax + b\) Straight line Fixed cost + unit cost
Polynomial \(f(x) = ax^2 + bx + c\) U-shaped or ∩-shaped Quadratic cost curve
Logarithm \(f(x) = \ln(x)\), \(x > 0\) Grows slowly Utility, elasticity
Exponential \(f(x) = e^x\) Grows rapidly Compound interest

3. Second-Degree Equations

General form: \(ax^2 + bx + c = 0\) with \(a \neq 0\)

Discriminant

\[\Delta = b^2 - 4ac\]

Case Number of real roots Solution(s)
\(\Delta > 0\) Two distinct roots \(x = \dfrac{-b \pm \sqrt{\Delta}}{2a}\)
\(\Delta = 0\) One repeated root \(x = \dfrac{-b}{2a}\)
\(\Delta < 0\) No real roots —

Worked Example — Choosing a Model

Scenario 1 — Sales commission

A salesperson earns €2,000 base salary plus €50 per unit sold.

\[f(x) = 50x + 2000 \quad \text{→ affine}\]

Scenario 2 — Factory cost with quadratic overhead

\[C(q) = 0.5q^2 + 10q + 500 \quad \text{→ polynomial}\]

Scenario 3 — Compound interest balance

\[B(t) = B_0\, e^{rt} \quad \text{→ exponential}\]

No calculation needed here — the goal is model identification.

Summary

  1. The domain excludes values that make the function undefined — always check for forbidden values (zero denominators, negative radicands).

  2. Affine = linear growth · Polynomial = curved · Logarithm = slow growth · Exponential = rapid growth.

  3. For a second-degree equation, compute \(\Delta = b^2 - 4ac\) first — it determines the number of real solutions.

📖 Full Reference

→ Module 6: Functions Useful in Economics and Management