EN112_Slide_Show_functions

Page 1: Introduction

• Course Title: EN112 - Engineering Mathematics I

• Department: Mathematics and Computer Science

• Institution: The Papua New Guinea University of Technology

• Semester: Semester 1, 2025

Page 2: Functions

• Definition: A function f is a rule that associates each input x to a unique output f(x).

• Functions can be expressed:

  • Numerically (tables)

  • Geometrically (graphs)

  • Algebraically (formulas)

• Illustrates independent (input) and dependent (output) variables with an example y = 1/x.

Page 3: Vertical Line Test

• A curve represents a function if a vertical line intersects it at most once.

• Example: The circle defined by x² + y² = 25 fails the test; thus, it's not a function of x.

• It can be viewed as two semi-circles: y = √(25 - x²) and y = -√(25 - x²).

Page 4: Domain and Range

• Domain: All input numbers x that a function can process.

• Range: The corresponding collection of output numbers y.

• Example: For y = √(1 - x²), the domain is -1 ≤ x ≤ 1, and the range is 0 ≤ y ≤ 1.

Page 5: Composition of Functions

• Functions can be chained; the output from one function becomes the input of another.

• Notation: f ◦ g ◦ h = f(g(h(x))).

• Example: For f(x) = √x, g(x) = 1/x, h(x) = x³, the composition yields (f ◦ g ◦ h)(x) = 1/x^(3/2).

Page 6: Inverse Functions

• Inverse functions reverse the operations of the original function, denoted f⁻¹(x).

• Examples:

  • f(x) = kx (linear)

  • f(x) = k/x (inverse proportional)

• Examples include Boyle's Law P = kV and Newtonian viscosity τx = µ ∂u/∂y.

Page 7: Polynomial Functions

• Common polynomial types include linear (y = mx + c), quadratic (y = ax² + bx + c), and cubic (y = x³).

• 2nd-order polynomials' roots can be found using the quadratic formula:

  • x = [-b ± √(b² - 4ac)] / 2a.

Page 8: Completing the Square

• Completed square form of a quadratic function is derived as follows:

  • (x + d)² = x² + 2dx + d².

• Example: For f(x) = x² + 6x - 3:

  • Completed to (x + 3)² - 12.

Page 9: Other Function Features

• Examples of cubic functions include f(x) = x³, f(x) = (x − 3)³, and others.

• Functions might also include absolute, square root, and trigonometric functions (SOH CAH TOA).

• Recall index laws for exponents.

Page 10: Exponential Functions

• Differences between polynomial (f(x) = xᵐ) and exponential functions (f(x) = m^x) are crucial for calculation speed.

• The Euler number e ≈ 2.718 is the basis for natural exponential functions.

• Derived from compound interest: e = lim(n→∞) (1 + 1/n)ⁿ.

Page 11: Logarithmic Functions

• The inverse of exponential functions, denoted log (base 10) or ln (base e).

• For example, log₁₀(100) = 2, ln(1) = 0.

• Change of base formula: logₐ x = ln x / ln a.

Page 12: Change of Logarithmic Base

• To solve (log₂81)(log₃32):

  • Use the identities from previous pages to simplify without a calculator, yielding a final value of 20.

Page 13: Change of Base Exercise

• Perform calculations for log values yielding 1.908485 / 0.30103 and 0.47712 resulting in a total of approximately 20.

Page 14: Radian Measure of Angles

• Defined via circumference-to-diameter ratio.

• 1 radian = 180/π degrees.

• Conversion formulas include x radians = (180/π) × x°.

Page 15: Trigonometric Functions

• Recap of trigonometric ratios: sine, cosine, and tangent (SOH CAH TOA).

• Inverse functions (arcsin, arccos, arctan) provide angle outputs.

• All functions are periodic, plotted with specific properties.

Page 16: Sinusoidal Functions

• General form: y = f(x) = Asin(Bx) or A*cos(Bx).

• Factors include amplitude (A), period (T = 2π/|B|), and frequency (f = |B|/2π).

Page 17: Generalized Sinusoidal Functions

• Extended form: y = Asin(Bx − C) or y = Acos(Bx − C), allowing phase shifts based on C.

• Relevant in wave studies, including frequency and bandwidth discussions.

Page 18: Trigonometric Identities

• Fundamental equations like sin²θ + cos²θ = 1 are essential for simplifying expressions.

• These identities are provable and necessary for operations in math.

Page 19: Hyperbolic Functions

• Defined as expressions involving e and e⁻.

• Similar properties to trigonometric functions including certain identities.

• Common hyperbolic functions: sinh, cosh, tanh.

Page 20: Inverse Hyperbolic Functions

• Defined in terms of natural logarithms:

  • For example, sinh⁻¹ x = ln(x + √(x² + 1)).

Page 21: Parametric Equations

• Functions expressed in terms of parameters (often time) for modeling.

• Example: Circle defined by x = cosθ, y = sinθ shows the relation x² + y² = 1.

Page 22: Parameterization in Hyperbolic Functions

• Expressed using parameters, resulting in curves described by x² − y² = 1.

• Example: x = cosh(t), y = sinh(t) represents a unit hyperbola.