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MCR3U Exponential Functions
Grade 11 Unit 3 — practice questions with full solutions

Negative and rational exponents, exponential graphs and asymptotes, and real growth and decay problems.

Free MCR3U exponential functions practice for Grade 11 students in Ontario. These 6 questions cover the same material as a typical unit test or exam review on this topic, and every one comes with a complete worked solution — not just an answer key. Useful whether you are preparing for a MCR3U unit test, catching up on a lesson, or reviewing before the final exam.
MCR3U · Grade 11 Exponential Functions 5 core 1 challenge
1Core
-3-2-11232468xy(0, 1)asymptote y = 0y = 2ˣy = (½)ˣ
Using the two curves shown: (a) state which is growth and which is decay, (b) give the equation of the horizontal asymptote, (c) explain why both pass through (0, 1), and (d) state the domain and range of y = 2ˣ.
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  1. (a) y = 2ˣ rises as x increases, so it models growth. y = (½)ˣ falls, so it models decay.
  2. (b) Both approach the x-axis without ever touching it, so the asymptote is y = 0.
  3. (c) Any non-zero base raised to the power 0 equals 1, so every basic exponential passes through (0, 1).
  4. (d) You can raise the base to any real power, so the domain is all real numbers; the output is always positive, so the range is y > 0.
Final answer(a) 2ˣ growth, (½)ˣ decay   (b) y = 0   (c) any base to the power 0 is 1   (d) domain ℝ, range y > 0
2Core
Evaluate without a calculator: (a) 163/4   (b) 27−2/3   (c) (4⁄9)−1/2
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  1. (a) The denominator of the exponent is the root: 4√16 = 2. Then raise to the power 3: 2³ = 8.
  2. (b) A negative exponent means reciprocal. First 272/3 = (3√27)² = 3² = 9, so 27−2/3 = 1⁄9.
  3. (c) The negative exponent flips the fraction: (9⁄4)1/2.
  4. Then take the square root of top and bottom: 3⁄2.
Final answer(a) 8   (b) 1⁄9   (c) 3⁄2
3Core
Simplify:   (2x³y⁻²)³ × (4x⁻¹y⁴)⁻¹, writing your answer with positive exponents only.
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  1. Cube the first bracket: 2³x⁹y⁻⁶ = 8x⁹y⁻⁶.
  2. Apply the −1 power to the second: 4⁻¹x¹y⁻⁴ = x⁄4y⁴.
  3. Multiply: 8x⁹y⁻⁶ × x ÷ (4y⁴) = 2x¹⁰y⁻¹⁰.
  4. Write with positive exponents by moving y to the denominator.
Final answer2x¹⁰⁄y¹⁰
4Core
Solve for x:   32x−1 = 81
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  1. Write both sides with the same base: 81 = 3⁴.
  2. So 32x−1 = 3⁴.
  3. With equal bases, the exponents must be equal: 2x − 1 = 4.
  4. Solve: 2x = 5, so x = 2.5.
Final answerx = 2.5
5Core
A bacterial culture doubles every 3 hours, starting from 500 bacteria. (a) Write an equation for the population after t hours. (b) Find the population after 12 hours. (c) How long until it reaches 8000?
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  1. (a) Doubling gives base 2, and the exponent is the number of doubling periods: P = 500(2)t/3.
  2. (b) At t = 12 there are 12 ÷ 3 = 4 doubling periods: P = 500(2)⁴ = 500 × 16 = 8000.
  3. (c) From part (b) the population is already 8000 at t = 12 hours.
  4. Confirm: 8000 ÷ 500 = 16 = 2⁴, so 4 doublings, and 4 × 3 = 12 hours.
Final answer(a) P = 500(2)t/3   (b) 8000   (c) 12 hours
6Challenge
A radioactive substance has a half-life of 8 days. A sample starts at 240 g. (a) Write the decay equation. (b) Find the mass after 20 days, to one decimal place.
Show the full solution
  1. (a) Half-life gives base ½, with the exponent counting half-life periods: M = 240(½)t/8.
  2. (b) At t = 20 the exponent is 20 ÷ 8 = 2.5.
  3. So M = 240 × (0.5)2.5.
  4. (0.5)2.5 ≈ 0.176777, so M ≈ 240 × 0.176777 ≈ 42.4 g.
Final answer(a) M = 240(½)t/8   (b) ≈ 42.4 g

MCR3U Exponential Functions — common questions

Short answers to the things students ask most about this unit.

What does a fractional exponent mean?

The denominator is the root and the numerator is the power. So 16^(3/4) means take the fourth root of 16 to get 2, then cube it to get 8.

How do you set up a half-life or doubling problem?

Use A = A₀ · b^(t/p), where b is 2 for doubling or ½ for half-life, and p is the length of one doubling or half-life period.

More MCR3U practice — grade 11 functions

MCR3U Functions & TransformationsMCR3U Quadratic FunctionsMCR3U Sequences & SeriesMCR3U Financial ApplicationsAll MCR3U units
← Unit 2: Quadratic FunctionsUnit 4: Sequences & Series →
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