Overview
This lecture covers numerical functions: their definition, domain (definition set), equality and operations on functions, function composition, direction of change, graphical representation, symmetry, and practical steps for second secondary mathematics.
Definition and Domain of Functions
- A function f associates each x in its definition set (DF) to a unique real number f(x).
- The domain (DF) includes all real values of x for which f(x) is defined.
- For square roots: the expression inside must be non-negative.
- For fractions: the denominator must not be zero.
- For absolute values: check when expressions become zero or undefined.
Determining the Domain: Practical Cases
- Functions with no roots or denominators: DF = ℝ.
- Denominator includes a variable: exclude values making denominator zero.
- Roots with variables: set the inside expression ≥ 0.
- Absolute value in the denominator: exclude values making it zero.
- Intersection of conditions is the domain for combined cases.
Equality of Functions
- Two functions f and g are equal iff: they have the same domain and f(x) = g(x) for all x in the domain.
Algebraic Operations on Functions
- Sum/difference: (f ± g)(x) = f(x) ± g(x); domain = intersection of f and g domains.
- Product: (f·g)(x) = f(x)·g(x); domain = intersection of domains.
- Scalar multiplication: (k·f)(x) = k·f(x); domain = domain of f.
- Quotient: (f/g)(x) = f(x)/g(x); domain = intersection of domains, excluding where g(x)=0.
Composition of Functions
- The composition (f∘g)(x) = f(g(x)).
- The domain of f∘g: all x in domain of g such that g(x) is in domain of f.
- Composition is not commutative: generally, f∘g ≠ g∘f.
Direction of Change (Monotonicity)
- If k > 0, then f(x) and k·f(x) have the same direction of change.
- If k < 0, the direction of change is reversed.
- The direction of f∘g depends on the monotonicity of both f and g.
Graphical Representation and Transformations
- f(x)+k: graph shifts up by k.
- f(x–a): graph shifts right by a.
- k·f(x): vertical stretch or reflection if k<0.
- f(–x): reflection over the y-axis.
- –f(x): reflection over the x-axis.
- |f(x)|: all values below x-axis reflected above.
- f(|x|): mirror the graph for x<0 across the y-axis.
Symmetry: Axis and Center
- Axis of symmetry x=a: f(2a–x) = f(x) for all x in domain.
- Center of symmetry (a, b): f(2a–x) + f(x) = 2b for all x in domain.
Key Terms & Definitions
- Function — A rule associating each input x in the domain with a unique output f(x).
- Domain (DF) — The set of all real numbers x where f(x) is defined.
- Image — The value f(x) corresponding to a specific x.
- Composition — The function formed by applying one function to the result of another: (f∘g)(x).
- Monotonic Function — A function that is either always increasing or always decreasing over its domain.
- Axis of symmetry — A line where the graph of the function mirrors itself.
- Center of symmetry — A point about which the graph is symmetric.
Action Items / Next Steps
- Review and summarize key examples of domain determination.
- Practice function operations and compositions using provided exercises.
- Study the table summarizing graphical transformations and symmetry.
- Attempt homework and review exercises from the recommended textbook.
- Watch detailed exercise explanations on the referenced YouTube channel.
- Prepare a table of change for given functions as practice.