W3_Functions and Graphs

Fundamentals of Mathematics 1 Session Overview

Title: Functions and GraphsInstructor: Erwan LamyInstitution: ESCP Business SchoolLocations: Berlin, London, Madrid, Paris, Turin, Warsaw

Objectives of the Session

• Understand the concept of functions.

• Manipulate functions: combine and find inverse functions.

• Graph equations and functions.

• Study symmetry with respect to: x-axis, y-axis, origin, line y=x.

• Analyze the signs of a function and solve nonlinear inequalities.

Session Outline

1. Functions (Chapter 2.1)

   • Methods: Definition and evaluation of functions, real-life examples of function applications.

2. Special Functions (Chapters 2.2, 3.2, 3.3)

   • Methods: Identify and differentiate among constant, linear, quadratic, polynomial, rational, case-defined, and absolute value functions using graphs and algebraic manipulation.

3. Applications (Chapter 2.1)

   • Methods: Use demand and supply functions; derive implications from these functions regarding price and quantity.

4. Combinations of Functions (Chapter 2.3)

   • Methods: Algebraic operations to combine functions (sum, product, quotient) and calculate composition of functions through substitution.

5. Inverse Functions (Chapter 2.4)

   • Methods: Finding inverses through algebraic manipulation; graphically verify the formation of inverse relationships.

6. Graphs in Rectangular Coordinates (Chapter 2.5)

   • Methods: Plot functions on Cartesian coordinates, identify and calculate intercepts, investigate behavior as x approaches ±∞.

7. Symmetry (Chapter 2.6)

   • Methods: Algebraic tests for symmetry (evaluating f(-x)), graphical representation to demonstrate symmetry about axes and lines.

8. Translations and Reflections (Chapter 2.7)

   • Methods: Determine behavior under transformations; apply shifts and reflections to function equations and their graphs.

9. Nonlinear Inequalities (Chapter 10.4)

   • Methods: Use sign charts, factorization, and interval testing to find solutions of inequalities, applying Bolzano's theorem for root estimation.

Understanding Functions (Chapter 2.1)

• Definition: A function transforms an independent variable (input) to a dependent variable (output).

• Example: If f(x) = 25 + 100, then x is the input.

• Notationally: For a function defined as f: x ⟶ y, x is the independent variable, and y is the dependent variable.

Function Characteristics

• Input and Output: Elements of sets can be real numbers; inputs cannot be assigned more than one output.

• Domain: Set containing all valid inputs.

• Range: Set of all possible outputs.

Domain and Range (Examples)

• If f(x) = x^2 - 1, the domain is all real numbers (ℝ).

• For function g(x) = √(x - 3), the domain is [3, ∞).

Function Equality (Chapter 2.2)

• Condition: Two functions f and g are equal if their domains are equal and for all x in the domain, f(x) = g(x).

Special Functions (Chapter 2.2)

• Constant Function: f(x) = c (where c is constant). Domain: ℝ.

• Linear Function: Can be written as f(x) = mx + b. Example: f(x) = 2x - 1.

• Quadratic Function: Can be represented as f(x) = ax^2 + bx + c (a ≠ 0). Example: f(x) = -2x^2 - 4x + 12.

• Polynomial Functions: Defined by: f(x) = a_n x^n + ... + a_1 x + a_0, where n is a nonnegative integer. Degree indicates the highest exponent of x. Example: f(x) = 4x^3 + 12x^2 - 8x + 9 (degree 3, leading coefficient 4).

• Rational Functions: Defined as the quotient of two polynomial functions. Example: f(x) = (x^2 - 3x + 5)/(x + 2).

• Case-Defined Functions: Use different formulas to define outputs over disjoint parts of the domain. Example: f(x) = {√x if 0 ≤ x < 1; 5 if 3 ≤ x < 5; 4 if 5 < x ≤ 7}.

• Absolute Value Function: f(x) = |x| = {x if x ≥ 0; -x if x < 0}.

Demand and Supply Functions (Applications) (Chapter 2.1)

• Demand Function: p(q) defines how price p varies with quantity q. Example: p(q) = 100 - 0.5q.

• Supply Function: p(q) describes the relationship between price p and the quantity q supplied. Typically, higher quantity = higher price.

Combinations of Functions (Chapter 2.3)

• Methods: Functions can be combined using operations:

  • Sum: (f + g)(x) = f(x) + g(x).

  • Product: (fg)(x) = f(x) * g(x).

  • Quotient: (f/g)(x) = f(x)/g(x).

Composition of Functions

• Definition: Composition is applying one function to the result of another.

• Denoted: (f ◦ g)(x) = f(g(x)).

• Example: If f(x) = x + 1 and g(x) = 2x, then (f ◦ g)(x) = f(g(x)) = 2x + 1.

Inverse Functions (Chapter 2.4)

• Definition: Inverses return the input from the output of a function.

• Denoted: f⁻¹(y) such that f(f⁻¹(y)) = y.

• Condition: A function must be one-to-one to have an inverse.

Graphs in Rectangular Coordinates (Chapter 2.5)

• Definition: The rectangular coordinate system is used to visualize functions by plotting points (x, f(x)).

• Intercepts: X-intercept is where y = 0; Y-intercept is where x = 0.

Symmetry Analysis (Chapter 2.6)

• Conditions: A graph may display:

  • Symmetry about the y-axis if f(-x) = f(x).

  • Symmetry about the x-axis if f(x) = -f(x).

  • Symmetry about the origin if f(-x) = -f(x).

  • Symmetry about the line y = x if f(x) = f⁻¹(x).

Transformations (Chapter 2.7)

• Methods: Transformations can alter the graph of a function:

  • Vertical shifts: Effect on y-values.

  • Horizontal shifts: Effect on x-values.

  • Reflections: Invert over x-axis or y-axis.

  • Stretches: Alter steepness and width.

Nonlinear Inequalities (Chapter 10.4)

• Methods: Nonlinear inequalities can be analyzed using sign charts.

• A function f(x) changing sign indicates a root in that interval (Bolzano's theorem).

• Solutions to inequalities: Can be obtained through factorization and analyzing intervals.

Fundamentals of Mathematics 1 Session Overview

Title: Functions and GraphsInstructor: Erwan LamyInstitution: ESCP Business SchoolLocations: Berlin, London, Madrid, Paris, Turin, Warsaw

Objectives of the Session

• Understand the concept of functions.

• Manipulate functions: combine and find inverse functions.

• Graph equations and functions.

• Study symmetry with respect to: x-axis, y-axis, origin, line y=x.

• Analyze the signs of a function and solve nonlinear inequalities.

Session Outline

1. Functions (Chapter 2.1)

   • Methods: Definition and evaluation of functions, real-life examples of function applications.

2. Special Functions (Chapters 2.2, 3.2, 3.3)

   • Methods: Identify and differentiate constant, linear, quadratic, polynomial, rational, case-defined, and absolute value functions using graphs and algebraic manipulation.

3. Applications (Chapter 2.1)

   • Methods: Use demand and supply functions; derive implications regarding price and quantity.

4. Combinations of Functions (Chapter 2.3)

   • Methods: Algebraic operations to combine functions (sum, product, quotient), calculate compositions through substitution.

5. Inverse Functions (Chapter 2.4)

   • Methods: Find inverses through algebraic manipulation and verify graphically.

6. Graphs in Rectangular Coordinates (Chapter 2.5)

   • Methods: Plot functions on Cartesian coordinates, calculate intercepts, analyze behavior as x approaches ±∞.

7. Symmetry (Chapter 2.6)

   • Methods: Algebraic tests for symmetry (evaluating f(-x)), represent symmetry graphically.

8. Translations and Reflections (Chapter 2.7)

   • Methods: Analyze behavior under transformations (shifts, reflections).

9. Nonlinear Inequalities (Chapter 10.4)

   • Methods: Apply sign charts, factorization, and interval testing to find solutions of inequalities, utilizing Bolzano's theorem for root estimation.

Understanding Functions (Chapter 2.1)

• Definition: A function transforms an independent variable (input) to a dependent variable (output).

• Example: If f(x) = 25 + 100, then x is the input.

• Notationally: For a function defined as f: x ⟶ y, x is the independent variable, and y is the dependent variable.

Function Characteristics

• Input and Output: Inputs cannot yield more than one output.

• Domain: All valid inputs.

• Range: All possible outputs.

Domain and Range (Examples)

• If f(x) = x^2 - 1, domain: ℝ.

• For g(x) = √(x - 3), domain: [3, ∞).

Function Equality (Chapter 2.2)

• Condition: Two functions f and g are equal if their domains are equal and f(x) = g(x) for all x in the domain.

Special Functions (Chapter 2.2)

• Constant Function: f(x) = c; Domain: ℝ.

• Linear Function: f(x) = mx + b.

• Quadratic Function: f(x) = ax^2 + bx + c (a ≠ 0).

• Polynomial Functions: f(x) = a_n x^n + ... + a_1 x + a_0.

• Rational Functions: Quotient of two polynomials: f(x) = (x^2 - 3x + 5)/(x + 2).

• Case-Defined Functions: Different formulas on disjoint parts of the domain.

• Absolute Value Function: f(x) = |x|.

Demand and Supply Functions (Applications) (Chapter 2.1)

• Demand Function: p(q) shows price variation with quantity.

• Supply Function: p(q) relates price to quantity supplied.

Combinations of Functions (Chapter 2.3)

• Methods: Combine using operations: Sum, Product, Quotient.

Composition of Functions

• Definition: Applying one function to another: (f ◦ g)(x) = f(g(x)).

Inverse Functions (Chapter 2.4)

• Definition: Inverses return input from output: f(f⁻¹(y)) = y.

Graphs in Rectangular Coordinates (Chapter 2.5)

• Definition: Visualize functions by plotting (x, f(x)).

Symmetry Analysis (Chapter 2.6)

• Conditions: Check for symmetry about axes and lines.

Transformations (Chapter 2.7)

• Methods: Analyze vertical/horizontal shifts, reflections, and stretches.

Nonlinear Inequalities (Chapter 10.4)

• Methods: Analyze using sign charts and factorization to find solutions.