Grade 11 Math Subjects — Complete Lesson Notes
School Year 2026–2027 | STEM Strand
Subjects covered:
General Mathematics (1st Semester)
Pre-Calculus (1st Semester)
Statistics and Probability (2nd Semester)
SUBJECT 1: GENERAL MATHEMATICS
Lesson 1: Functions and Relations
Relation – a set of ordered pairs (x, y).
Function – a relation where every x has exactly one y.
Vertical Line Test: if a vertical line touches the graph more than once, it’s
NOT a function.
Domain (set of all x-values), Range (set of all y-values).
Representations: table, mapping diagram, graph, equation.
Lesson 2: Evaluating Functions
Substitute the given value of x into f(x) and simplify.
Example: f(x) = 2x + 3, find f(4) → f(4) = 2(4)+3 = 11
Lesson 3: Operations on Functions
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x)
(f · g)(x) = f(x) · g(x)
(f / g)(x) = f(x) / g(x), g(x) ≠ 0
Composition: (f∘g)(x) = f(g(x))
Lesson 4: Rational Functions, Equations, and Inequalities
Rational equation – contains a fraction with variable in denominator; solve by
cross-multiplying / LCD.
Rational inequality – solved using sign analysis (test points between critical
values).
, Rational function – f(x) = P(x)/Q(x); graphed with asymptotes.
Lesson 5: Graphing Rational Functions
Find domain (exclude zeros of denominator).
Vertical asymptotes: zeros of denominator (not cancelled).
Horizontal asymptote: compare degrees of numerator (n) and denominator
(m):
N<m→y=0
N = m → y = leading coeff ratio
N > m → no horizontal asymptote (may have oblique asymptote)
Find x- and y-intercepts, then plot/sketch.
Lesson 6: One-to-One and Inverse Functions
A function is one-to-one if no two x-values give the same y (Horizontal Line
Test).
Only one-to-one functions have inverses.
To find f⁻¹(x): switch x and y, solve for y.
Graph of f⁻¹ reflects f across y = x.
Lesson 7: Exponential Functions, Equations, and Inequalities
Form: f(x) = abˣ
Exponential equation: same base method, or use logarithms.
Exponential inequality: compare exponents (careful with base between 0 and
1 — flips inequality).
Applications: compound interest, population/bacteria growth, radioactive
decay (half-life).
Lesson 8: Logarithmic Functions, Equations, and Inequalities
Logₐx = y ⟺ aʸ = x
Laws of Logarithms:
Log(mn) = log m + log n
Log(m/n) = log m – log n
School Year 2026–2027 | STEM Strand
Subjects covered:
General Mathematics (1st Semester)
Pre-Calculus (1st Semester)
Statistics and Probability (2nd Semester)
SUBJECT 1: GENERAL MATHEMATICS
Lesson 1: Functions and Relations
Relation – a set of ordered pairs (x, y).
Function – a relation where every x has exactly one y.
Vertical Line Test: if a vertical line touches the graph more than once, it’s
NOT a function.
Domain (set of all x-values), Range (set of all y-values).
Representations: table, mapping diagram, graph, equation.
Lesson 2: Evaluating Functions
Substitute the given value of x into f(x) and simplify.
Example: f(x) = 2x + 3, find f(4) → f(4) = 2(4)+3 = 11
Lesson 3: Operations on Functions
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x)
(f · g)(x) = f(x) · g(x)
(f / g)(x) = f(x) / g(x), g(x) ≠ 0
Composition: (f∘g)(x) = f(g(x))
Lesson 4: Rational Functions, Equations, and Inequalities
Rational equation – contains a fraction with variable in denominator; solve by
cross-multiplying / LCD.
Rational inequality – solved using sign analysis (test points between critical
values).
, Rational function – f(x) = P(x)/Q(x); graphed with asymptotes.
Lesson 5: Graphing Rational Functions
Find domain (exclude zeros of denominator).
Vertical asymptotes: zeros of denominator (not cancelled).
Horizontal asymptote: compare degrees of numerator (n) and denominator
(m):
N<m→y=0
N = m → y = leading coeff ratio
N > m → no horizontal asymptote (may have oblique asymptote)
Find x- and y-intercepts, then plot/sketch.
Lesson 6: One-to-One and Inverse Functions
A function is one-to-one if no two x-values give the same y (Horizontal Line
Test).
Only one-to-one functions have inverses.
To find f⁻¹(x): switch x and y, solve for y.
Graph of f⁻¹ reflects f across y = x.
Lesson 7: Exponential Functions, Equations, and Inequalities
Form: f(x) = abˣ
Exponential equation: same base method, or use logarithms.
Exponential inequality: compare exponents (careful with base between 0 and
1 — flips inequality).
Applications: compound interest, population/bacteria growth, radioactive
decay (half-life).
Lesson 8: Logarithmic Functions, Equations, and Inequalities
Logₐx = y ⟺ aʸ = x
Laws of Logarithms:
Log(mn) = log m + log n
Log(m/n) = log m – log n
