Functions

Checkpoints

• Elementary Functions Basis Surveys the complete family of elementary functions — polynomials, rational and algebraic functions, exponentials, logarithms, trigonometric, inverse trigonometric, and hyperbolic functions — and defines precisely what it means for a function to be elementary through finite combinations of arithmetic operations and composition.
• Exponential Functions Basis Defines the exponential function exp(x) via its absolutely convergent power series ∑ xᵏ/k!, proves the functional equation exp(x+y) = exp(x)exp(y) using the Cauchy product, and establishes that exp is strictly positive, strictly increasing, its own derivative, and surjects onto (0, ∞); also defines general exponentials bˣ and shows why e is the uniquely natural base.
• Hyperbolic Functions and Their Inverses Basis Defines the hyperbolic functions sinh, cosh, and tanh from the exponential function, establishes their fundamental identity, addition formulas, and derivatives, then derives the inverse hyperbolic functions as closed-form logarithmic expressions.
• Inverse Trigonometric Functions Basis Defines the six inverse trigonometric functions by restricting each trigonometric function to an interval where it becomes injective, and derives their derivatives using the inverse function theorem along with key identities such as arcsin x + arccos x = π/2.
• Logarithms Basis Defines the natural logarithm as the inverse of the exponential function and derives its key properties — the product rule, power rule, and derivative — then extends the framework to logarithms and exponentials in any base.
• Polynomial Functions Basis Defines polynomial functions over ℝ, develops their algebra as the ring ℝ[x], establishes the division algorithm with remainder and factor theorems, classifies roots by multiplicity, states the Fundamental Theorem of Algebra, and shows how every real polynomial factors into linear and irreducible quadratic pieces.
• Trigonometric Functions Basis Defines sine and cosine rigorously as power series, derives their derivatives, the Pythagorean identity, and the addition formulas, uses these to define π analytically, and introduces the full family of trigonometric functions with their derivatives.