Calculus I: Functions and Function Composition

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.

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.

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.

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.

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).

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.

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.