MCR3U Grade 11 Functions: Topics & Practice Questions | IntelliX

MCR3U is Ontario’s Grade 11 university-preparation Functions course. It develops function reasoning from earlier algebra and graphs. Its standard prerequisite is MPM2D. This guide explains what to review and provides concrete worked examples and readiness questions without relying on memorised steps alone.

What Topics Does MCR3U Cover?

The curriculum is structured around four core strands: characteristics of functions, exponential functions, discrete functions (sequences, series, and financial math), and trigonometric functions. Students connect equations, tables and graphs, evaluate transformations, determine inverses, and apply functions to real-world modeling.

What to Review Before MCR3U: Readiness Foundations

Before starting MCR3U, students must be confident with quadratic factoring, exponent rules, and solving linear systems. If quadratics feel shaky, dedicate time to reviewing factoring techniques and vertex forms.

Understanding Function Notation: f(x) vs. Solving f(x)

If f(x) = 2x + 3, evaluating f(4) asks for the output when input x = 4: f(4) = 2(4) + 3 = 11. It does NOT mean 'f multiplied by 4'. In contrast, solving f(x) = 11 asks for the unknown input that yields 11: 2x + 3 = 11 leads to x = 4. Distinguishing inputs from outputs prevents confusion across all four units.

Worked Example 1: Finding an Inverse Function & Stating Domain

Problem: Given f(x) = 3x − 7, determine f⁻¹(x) and state the domain and range of both f and f⁻¹.

Step 1: Replace f(x) with y: y = 3x − 7. Step 2: Swap variables x and y to represent inverse reflection across the line y = x: x = 3y − 7. Step 3: Solve for y: x + 7 = 3y, so y = (x + 7) / 3. Step 4: Express in inverse notation: f⁻¹(x) = (x + 7) / 3. Domain and Range: Since f(x) is linear, Domain(f) = {x ∈ ℝ}, Range(f) = {y ∈ ℝ}. Correspondingly, Domain(f⁻¹) = {x ∈ ℝ} and Range(f⁻¹) = {y ∈ ℝ}.

Worked Example 2: Exponential Transformation & Asymptote

Problem: Given g(x) = 2(3)^(x − 1) + 4, identify the base function, list all transformations, state the horizontal asymptote, and evaluate g(3).

Base Function: y = 3^x. Transformations: Vertical stretch by factor of 2, horizontal translation 1 unit to the right, and vertical translation 4 units up. Horizontal Asymptote: The standard base asymptote y = 0 shifts up by 4, giving y = 4. Evaluation: g(3) = 2(3)^(3 − 1) + 4 = 2(3^2) + 4 = 2(9) + 4 = 18 + 4 = 22.

Worked Example 3: Trigonometry — Exact Values with Special Triangles

Problem: Evaluate the exact value of sin(150°) and cos(150°) without a calculator.

Step 1: Determine the reference angle. 150° is in Quadrant II. Reference angle θ_r = 180° − 150° = 30°. Step 2: Use the special 30°-60°-90° triangle where opposite = 1, adjacent = √3, hypotenuse = 2. Step 3: In Quadrant II, sine is positive (CAST rule) and cosine is negative. Conclusion: sin(150°) = sin(30°) = 1/2. cos(150°) = −cos(30°) = −√3/2.

Common Student Errors in MCR3U

• Confusing f⁻¹(x) with 1/f(x): The -1 exponent denotes inverse function reflection, NOT reciprocal power.
• Distributing Exponents Over Addition: Writing (x + y)² as x² + y² instead of x² + 2xy + y².
• Ignoring Asymptotes in Domain/Range: Claiming y > 0 for y = 2^x - 3 instead of y > -3.
• Sign Errors in the CAST Rule: Forgetting that cos(θ) and tan(θ) are negative in Quadrant II.
• Omitting Excluded Values: Failing to record denominator restrictions before cancelling rational terms.

Five Diagnostic Readiness Questions (with Solutions)

• Q1: Factor 2x² - 5x - 3 completely. Answer: (2x + 1)(x - 3).
• Q2: Simplify (x²y³)^(-2). Answer: 1 / (x⁴y⁶).
• Q3: If h(x) = x² - 4, what is h(-3)? Answer: (-3)² - 4 = 9 - 4 = 5 (note that negative squared is positive).
• Q4: In a right triangle with hypotenuse 10 and adjacent side 6, find the cosine of the angle. Answer: cos(θ) = 6/10 = 3/5.
• Q5: Solve 3^(2x) = 27. Answer: 3^(2x) = 3^3 implies 2x = 3, so x = 3/2 or 1.5.

Connecting MCR3U to Senior Mathematics

MCR3U is one prerequisite route into senior mathematics. Ontario also lists MCT4C for MHF4U and MCF3M for MDM4U. Check the exact destination course and discuss the sequence with guidance.

Check whether a relation is a function

Consider the pairs (1, 3), (2, 5), (1, 7). This relation is not a function because input 1 has two different outputs. By contrast, (1, 3), (2, 3), (3, 3) is a function: different inputs may share an output. Use this distinction alongside the vertical-line test, then explain why swapping inputs and outputs may produce an inverse relation that is not a function.

Domain, range and transformations: three explained checks

• For f(x) = √(x − 2), the radicand must be non-negative, so the domain is x ≥ 2. The principal square root is non-negative, so the range is y ≥ 0.
• For g(x) = −2(x − 1)² + 3, the graph opens downward with vertex (1, 3). Its domain is all real numbers and its range is y ≤ 3.
• For h(x) = 2^x − 4, the graph has horizontal asymptote y = −4. Its domain is all real numbers and its range is y > −4.