MCR3U Unit 1 Guide: Inverses, Domain, Range & Function Notation [With Worked Solutions]

MCR3U Unit 1 establishes foundational function reasoning in Ontario's Grade 11 university mathematics. Students transition from Grade 10 quadratic algebra to formal function notation f(x), algebraic inverse determination via variable swapping, domain and range interval restrictions, and composite transformations y = af(k(x - d)) + c.

Official Ontario Curriculum Breakdown: MCR3U

• Ministry Course Code: MCR3U (Functions, Grade 11, University Preparation)
• Strand / Expectation: Characteristics of Functions (A1: Function representation, A2: Inverses and transformations)
• Prerequisite Course: MPM2D (Grade 10 Principles of Mathematics, 70%+ strongly recommended)
• Course Weighting: 70% Term Evaluations + 30% Final Culminating Task / Exam
• Post-Secondary Impact: Mandatory gateway prerequisite for Grade 12 Advanced Functions (MHF4U) and Calculus (MCV4U)

Worked Example 1: Finding the Inverse of a Rational Function with Domain/Range Restrictions

Problem: Given the rational function f(x) = (2x + 1) / (x - 3), find the equation for the inverse function f⁻¹(x), and state the domain and range of both f and f⁻¹.

• Step 1 (State restrictions on f): Denominator cannot equal zero: x - 3 ≠ 0 ⇒ x ≠ 3. Horizontal asymptote is y = 2/1 = 2 ⇒ y ≠ 2. Therefore, Domain(f) = {x ∈ ℝ | x ≠ 3}, Range(f) = {y ∈ ℝ | y ≠ 2}.
• Step 2 (Swap x and y to invert): Replace f(x) with y: y = (2x + 1)/(x - 3). Swap x and y: x = (2y + 1)/(y - 3).
• Step 3 (Clear denominator): Multiply both sides by (y - 3): x(y - 3) = 2y + 1 ⇒ xy - 3x = 2y + 1.
• Step 4 (Collect y-terms on one side): xy - 2y = 3x + 1.
• Step 5 (Factor out y and isolate): y(x - 2) = 3x + 1 ⇒ y = (3x + 1) / (x - 2).
• Step 6 (Inverse Notation & Domains): f⁻¹(x) = (3x + 1) / (x - 2). Domain(f⁻¹) = Range(f) = {x ∈ ℝ | x ≠ 2}; Range(f⁻¹) = Domain(f) = {y ∈ ℝ | y ≠ 3}.

Teacher's Strategy: Always verify the inverse relationship: Domain of the original function permanently becomes the Range of the inverse function, and Range of original becomes Domain of inverse.

Worked Example 2: Applying the Universal Transformation Mapping Rule to a Square Root Function

Problem: Describe the transformations applied to the parent function f(x) = √x to obtain g(x) = -2√(3x - 12) + 5, state the mapping rule, and find the transformed coordinates of key point (4, 2).

• Step 1 (Factor out k from inside the argument): g(x) = -2√[3(x - 4)] + 5. (Critical: k = 3 must be factored out to find the true horizontal translation).
• Step 2 (Identify transformation parameters): a = -2 (vertical reflection in x-axis, vertical stretch by factor of 2); k = 3 (horizontal compression by factor of 1/3); d = +4 (horizontal shift 4 units right); c = +5 (vertical shift 5 units up).
• Step 3 (Formulate the coordinate mapping rule): (x, y) → (x/3 + 4, -2y + 5).
• Step 4 (Transform point (4, 2)): New x = (4/3) + 4 = 16/3 ≈ 5.33. New y = -2(2) + 5 = 1. Transformed point is (16/3, 1).

Teacher's Strategy: Ontario unit tests heavily penalize students who forget to factor out k in expressions like √(3x - 12). If you do not factor, your horizontal shift will incorrectly be 12 instead of 4.

The 5 Most Common Mistakes Ontario Students Make in MCR3U

• Confusing inverse notation f⁻¹(x) with reciprocal 1/f(x). Inverse means reflecting across y = x; reciprocal means 1 divided by the output.
• Failing to factor out 'k' from transformed functions like f(2x + 6), which leads to wrong horizontal translations.
• Omitting restricted values from domain statements when rational expressions cancel out.
• Incorrectly claiming the range of y = √(x - 3) includes negative numbers (the principal square root is strictly non-negative: y ≥ 0).
• Confusing horizontal stretch/compression factors (k = 2 compresses horizontally by 1/2; k = 1/2 stretches horizontally by 2).

University Admissions & Prerequisite Cutoff Impact

MCR3U is evaluated by Ontario universities during early rolling admission rounds (December through March) as a direct predictor of senior math capability. Students with marks below 75% in MCR3U face significant hurdles in Grade 12 Advanced Functions (MHF4U), which directly determines STEM, Computer Science, and Business degree eligibility.

Diagnostic Readiness Check: Test Your Knowledge

• What is the domain of f(x) = 1 / √(x - 5)? — Answer: {x ∈ ℝ | x > 5} (radicand must be strictly positive because division by zero is undefined).
• If (a, b) is on f(x), what point must be on f⁻¹(x)? — Answer: (b, a) due to reflection over the diagonal line y = x.
• How does the graph of y = f(-x) differ from y = -f(x)? — Answer: y = f(-x) is a reflection across the y-axis; y = -f(x) is a reflection across the x-axis.