Overview
This lecture covers numerical functions: definitions, domain, operations (sum, product, quotient), function composition, direction of change (monotonicity), graphical transformations, and symmetry (axis and center), which are major topics in second-year secondary mathematics.
Numerical Function Definitions & Domains
- A numerical function assigns each real number x in its domain (DF) to a unique real number f(x).
- The domain (DF) is the set of all real x for which f(x) is defined.
- Common domain restrictions are: denominator ≠ 0, expression under square root ≥ 0, and denominator under root ≠ 0.
Finding the Domain
- For polynomial functions: domain is all real numbers.
- For rational functions: exclude values that make the denominator zero.
- For functions involving square roots: require the expression under the root to be ≥ 0.
- For functions involving absolute value in the denominator: denominator ≠ 0.
Equality and Operations on Functions
- Two functions are equal if they have the same domain and equal expressions for all x in the domain.
- Sum/Difference/Product: domain is the intersection of the domains of the involved functions.
- Quotient: domain is intersection of domains, but exclude zeros of the denominator.
Composition of Functions
- The composite function f∘g is defined for x in the domain of g, where g(x) is in the domain of f.
- To find the domain of f∘g: x must be in domain of g and g(x) in domain of f.
- Composition is not commutative: f∘g ≠ g∘f in general.
Direction of Change (Monotonicity)
- A function is monotonic if it is either always increasing or decreasing on its domain.
- Multiplying a function by a positive constant retains the direction of change; by a negative, it reverses it.
- The composition's monotonicity depends on those of the component functions.
Graphical Transformations
- Adding a constant: f(x) + k shifts the graph vertically.
- f(x + b): shifts the graph horizontally.
- Multiplying output by λ > 0 stretches/compresses vertically; λ < 0 reflects over the x-axis.
- f(-x) reflects over the y-axis.
Symmetry: Axis and Center
- Axis of symmetry: vertical line x = a where f(a - x) = f(a + x) (even function about x = a).
- Center of symmetry: point (a, b) where f(a + x) + f(a - x) = 2b (odd function about (a, b)).
- Three methods for verifying symmetry: change of variable, direct substitution, algebraic manipulation.
Key Terms & Definitions
- Domain (DF) — Set of all real x for which f(x) is defined.
- Composite Function — Function formed by applying one function to the result of another: f∘g(x) = f(g(x)).
- Monotonic Function — A function that is either entirely non-increasing or non-decreasing over its domain.
- Axis of Symmetry — Vertical line where function graph is mirrored (x = a).
- Center of Symmetry — Point about which a graph is symmetric (a, b).
Action Items / Next Steps
- Review detailed exercises and solved examples in the recommended textbook (Silver Series Second Edition, page references given in lecture).
- Practice extracting the domain for various types of functions.
- Construct change tables (monotonicity tables) and practice identifying symmetry.
- Complete assigned homework and watch recommended YouTube solution videos for additional clarification.