calculus-notes-week04

Arithmetic Composition Inverses Exponentials Logarithms Exercises

Week 4: Composition, Inverses, Logarithms

• MA161/MA1161: Semester 1 Calculus

• Prof. Götz Pfeiffer, School of Mathematical and Statistical Sciences, University of Galway, October 7-8, 2024

New Functions from Old

• New functions can be derived from existing functions through:

  • Shifting:

    • g(x) = f(x) ± c

    • g(x) = f(x ± c)

  • Stretching:

    • g(x) = c f(x)

    • g(x) = f(c x), where c > 0

  • Reflecting:

    • g(x) = −f(x)

    • g(x) = f(−x)

    • g(x) = −f(−x)

• Important for sketching graphs quickly.

• Functions f(x) and g(x) can be combined using arithmetic operations to form h(x).

• Composition of functions is defined as f ◦ g (applying g first, then f).

Sums and Products of Functions

• Defined by

  • (f + g)(x) = f(x) + g(x)

  • (f - g)(x) = f(x) - g(x)

  • (f · g)(x) = f(x) · g(x)

  • (f/g)(x) = f(x)/g(x)

• Domains:

  • f + g, f - g, f · g: Intersection A ∩ B of domains.

  • f/g: {x ∈ A ∩ B | g(x) ≠ 0}

• Example with domains and functions f(x) = √x and g(x) = √(2−x).

Composition of Functions

• If y = f(u) and u = g(x), then y can be expressed as:

  • y = f(g(x)) = √(x² + 1)

• Composite function notation: (f ◦ g)(x) = f(g(x))

• "F of g of x" means applying g first, then f.

Examples of Composites

• Example: For f(x) = x² and g(x) = x - 3,

  • Compute (f ◦ g)(x) = f(g(x)) = (x - 3)²

  • Compute (g ◦ f)(x) = g(f(x)) = x² - 3

• Note that generally, f ◦ g ≠ g ◦ f

• Possible to compose multiple functions, e.g., (f ◦ g ◦ h)(x)

• Example composition found for F(x) = cos²(x + 9).

One-to-One Functions

• If f : D → R are functions of time (t) and population sizes (N):

• One-to-one function defined as:

  • f(x1) ≠ f(x2) when x1 ≠ x2

• Horizontal line test:

  • A function is one-to-one if it passes this test.

Horizontal Line Test

• Example for f(x) = x³ (one-to-one) vs. g(x) = x² (not one-to-one).

Inverse Functions

• For one-to-one functions, inverse f⁻¹ has domain B and range A,

• Defined by f⁻¹(x) = y ⇐⇒ f(y) = x.

• Distinction: f⁻¹(x) vs. reciprocal f(x)⁻¹.

• Example to find inverse of f(x) = x³ + 2:

  • Rearranging leads to f⁻¹(x) = ∛(x - 2).

Exponential Functions

• f(x) = 2^x is an exponential function; variable x is the exponent.

• Distinct from power functions.

• Rapid growth characteristic of exponential functions (doubling values).

Properties of The Exponential Function f(x) = a^x

• f(0) = 1

• f manupulations and laws of exponents detailed (e.g., a^x * a^y = a^(x+y)).

Applications: Bacterial Growth

• For populations doubling in controlled environment, model:

  • p(t) = 1000 · 2^(t/6)

Applications: Radioactive Decay

• Model mass after time t, m(t) = 24 · 2^(-t/25).

• Calculation of remaining mass after specific times.

Logarithmic Functions

• y = log_a(x) inversely relates to exponential ax.

• e.g., log_2(8) = 3 since 2^3 = 8.

Properties of Logarithmic Functions

• Cancellation properties and laws of logarithms detailed (log_a(xy) = log_a(x) + log_a(y))

The Natural Logarithm

• Defined as ln x = log_e(x), where e ≈ 2.718.

• Change of Base formula: log_a(x) = ln(x) / ln(a).

Applications

• Solve equations involving logarithms and model bacterial growth transformation for certain conditions.

Exercises

• A series of practical exercises are presented for the student, covering shifts, transformations, and properties of functions.