Calculus I: Functions and Function Composition

Overview

• In Calculus I, focus is on the idea of functions and how they act as rules that map inputs to outputs.
• The transcript mentions FFG and how it acts, which suggests function composition (often denoted f ∘ g). We’ll cover what a composite function is and how it behaves.
• The session ends with a plan to start working on function questions, i.e., some practice problems to apply these concepts.

Core Concepts

• What is a function?
  • A function f from a set A (domain) to a set B (codomain) assigns to each input a ∈ A exactly one output b ∈ B.
  • Notation example: $f: A \to B, \quad f(a) = b$
  • Domain and codomain:
  • Domain: the set of permissible inputs for which the rule is defined.
  • Codomain: the set from which outputs are drawn (may or may not be the same as the range).
  • Range (or image): the set of actual outputs produced by the function, i.e., $\text{Range}(f) = { f(a) \mid a \in \text{Dom}(f) }$
• Function notation and graphs
  • Functions are typically written as f(x) or g(x) to denote the rule applied to input x.
  • The graph of a function f is the collection of points (a, f(a)) corresponding to inputs in the domain.
• Function operations (brief intro)
  • Pointwise addition/subtraction: if f and g are defined on the same domain, then (f + g)(x) = f(x) + g(x).
  • Function composition (the focus for "FFG"):
  • Definition: $(f \circ g)(x) = f(g(x))$
  • Order matters: g first, then f.
  • Domain of the composition: the set of x for which x ∈ Dom(g) and g(x) ∈ Dom(f).
  • Significance: a core operation in calculus (chain rule, nested functions, etc.).
• Why this matters in calculus
  • Understanding how rules combine helps in analyzing more complex functions and in applying the chain rule for derivatives.

Function Notation and Domains

• Formal notes:
  • A function is often written as f: A → B with Dom(f) ⊆ A and codomain B.
  • For any a ∈ Dom(f), the output is f(a) ∈ B.
• Important definitions:
  • Domain: the set of all inputs for which the function rule is defined.
  • Range: the set of all actual outputs produced by the inputs in the domain.
• Examples:
  • If f(x) = x^2 and g(x) = x + 1, the composite is (f ∘ g)(x) = f(g(x)) = (x + 1)^2.
  • If f(x) = √x (defined for x ≥ 0) and g(x) = x − 2, then (f ∘ g)(x) = √(x − 2). The domain of this composition is x ≥ 2.

Function Composition (FFG)

• Core formula:
  • $(f \circ g)(x) = f(g(x))$
• Domain considerations:
  • Domain of f ∘ g = { x ∈ Dom(g) | g(x) ∈ Dom(f) }.
• Simple illustrative examples:
  • Example 1: Let $f(x) = x^2$ and $g(x) = x + 3$. Then $(f \circ g)(x) = (x + 3)^2.$
  • Example 2: Let $f(x) = \sqrt{x}$ with domain x ≥ 0, and $g(x) = x - 4$. Then $(f \circ g)(x) = \sqrt{x - 4}$ with domain x ≥ 4.
• Common pitfalls
  • The domain can shrink after composition due to the inner output needing to be valid for the outer function.
  • Changing the order (g ∘ f) generally changes the result and domain.
• Practical significance
  • Composition is essential for applying the chain rule in differentiation and for composing multiple rules into a single overall function.

How to Analyze Functions

• Practical steps when given a function:
  • Identify the rule of the function and its domain.
  • Determine the range of the function, if possible.
  • Consider how the function behaves under composition and how domains intersect.
  • Use function notation to express outputs for given inputs and to express composed functions clearly.
• Real-world relevance
  • Functions model real-world processes where an input (independent variable) maps to an output (dependent variable).
  • Composition represents sequential processes (one process feeding into another).

Practice: Function Questions (from transcript cue)

• Problem 1: Let $f(x) = x^2$ and $g(x) = x + 3$. Compute the composition $(f \circ g)(x)$.
  • Solution: $(f \circ g)(x) = (x + 3)^2.$
• Problem 2: Let $f(x) = \sqrt{x}$ (domain x ≥ 0) and $g(x) = x - 4$. Determine the domain of $(f \circ g)(x) = \sqrt{x - 4}$.
  • Solution: The domain is $x ≥ 4.$
• Problem 3: Given functions $f: \mathbb{R} \to \mathbb{R}, f(x) = 2x + 1$ and $g: \mathbb{R} \to \mathbb{R}, g(x) = x^2$, determine both compositions and their domains.
  • Solutions:
  • $(f \circ g)(x) = f(g(x)) = 2x^2 + 1$ with domain $\mathbb{R}$.
  • $(g \circ f)(x) = g(f(x)) = (2x + 1)^2$ with domain $\mathbb{R}$.
• Note: These prompts align with the transcript's intention to start working on function questions and practice applying the concept of function composition.

Summary of Key Takeaways

• A function f maps inputs to outputs with a specified domain and codomain.
• Notation f(x) expresses the rule; the domain and range determine where the function is defined and what values it can take.
• Function composition, denoted (f ∘ g), applies g first, then f: $(f \circ g)(x) = f(g(x))$.
• The domain of a composition depends on both the domain of g and the domain of f applied to g(x).
• Analyzing functions involves understanding inputs, outputs, and how combining functions changes the domain and outputs; practice with function questions reinforces these concepts.