Grade 11 Functions (MCR3U)
MCR3U introduces the formal language of functions — a conceptual framework that organises all of the mathematics from here to university. You will study four families of functions: general function concepts, exponential, discrete (sequences), and trigonometric. Each family illuminates a different type of real-world behaviour, from population growth to sound waves.
1. Characteristics of Functions
Definition: Function
A function is a relation in which each element of the domain (input) corresponds to exactly one element of the range (output). We write $f: x \mapsto f(x)$, or $y = f(x)$.
Vertical Line Test: A graph represents a function if and only if every vertical line intersects the graph at most once.
Domain and Range
The domain is the set of all allowable inputs. The range is the set of all resulting outputs. When stating domain and range:
- Exclude values that make a denominator zero.
- Exclude values that make a radicand negative (for even roots).
- Exclude values that make a logarithm argument non-positive.
Worked Example 1.1 — Domain and Range
State the domain and range of $f(x) = \dfrac{2}{\sqrt{x - 3}}$.
Domain We need $x - 3 > 0$ (strict inequality because the square root is in the denominator). So $x > 3$, i.e., domain is $(3, \infty)$.
Range When $x$ is just above 3, $\sqrt{x-3} \to 0^+$ so $f(x) \to +\infty$. As $x \to \infty$, $f(x) \to 0^+$. So the range is $(0, \infty)$.
Transformations of Functions
Transformation Summary
Starting from the base function $y = f(x)$:
- $y = f(x) + k$: vertical translation up $k$ units
- $y = f(x - h)$: horizontal translation right $h$ units
- $y = af(x)$: vertical stretch by factor $|a|$; reflect in $x$-axis if $a < 0$
- $y = f(kx)$: horizontal compression by factor $\dfrac{1}{|k|}$; reflect in $y$-axis if $k < 0$
- $y = -f(x)$: reflection in the $x$-axis
- $y = f(-x)$: reflection in the $y$-axis
Worked Example 1.2 — Applying Transformations
Describe the transformations that map $y = \sqrt{x}$ to $y = -2\sqrt{x + 4} + 1$.
- $x \to x + 4$: horizontal translation 4 units to the left
- $\times (-2)$: vertical stretch by factor 2 and reflection in the $x$-axis
- $+ 1$: vertical translation 1 unit up
Applying in order: the vertex of the transformed curve is at $(-4, 1)$, opening downward.
Inverse Functions
The inverse of $f$, written $f^{-1}$, reverses the mapping: if $f(a) = b$ then $f^{-1}(b) = a$. To find the inverse algebraically, swap $x$ and $y$ and solve for $y$. The graphs of $f$ and $f^{-1}$ are reflections of each other in the line $y = x$.
Worked Example 1.3 — Finding an Inverse
Find the inverse of $f(x) = 3x - 7$ and verify.
Step 1 Write $y = 3x - 7$, then swap: $x = 3y - 7$.
Step 2 Solve for $y$: $y = \dfrac{x + 7}{3}$, so $f^{-1}(x) = \dfrac{x+7}{3}$.
Verify $f(f^{-1}(x)) = 3 \cdot \dfrac{x+7}{3} - 7 = x + 7 - 7 = x$. ✓
2. Exponential Functions
An exponential function has the form $f(x) = a \cdot b^x$ where $a \neq 0$, $b > 0$, and $b \neq 1$. These functions model growth (when $b > 1$) and decay (when $0 < b < 1$) processes such as population change, radioactive decay, and compound interest.
Key Properties of $f(x) = b^x$
- Domain: $\mathbb{R}$ (all real numbers)
- Range: $(0, \infty)$ (always positive)
- $y$-intercept: $(0, 1)$ — since $b^0 = 1$
- Horizontal asymptote: $y = 0$ (the $x$-axis)
- If $b > 1$: increasing function (exponential growth)
- If $0 < b < 1$: decreasing function (exponential decay)
Exponential Growth and Decay Models
Growth and Decay Formulas
Exponential growth: $A(t) = A_0 \cdot b^t$ or $A(t) = A_0 \cdot (1 + r)^t$
Half-life (decay): $A(t) = A_0 \cdot \left(\dfrac{1}{2}\right)^{t/h}$ where $h$ is the half-life
Doubling time (growth): $A(t) = A_0 \cdot 2^{t/d}$ where $d$ is the doubling time
Worked Example 2.1 — Radioactive Decay
Carbon-14 has a half-life of 5730 years. A sample initially contains 200 mg. How much remains after 17 190 years?
Step 1 Determine the number of half-lives: $\dfrac{17190}{5730} = 3$ half-lives.
Step 2 Apply the formula: $$A = 200 \cdot \left(\frac{1}{2}\right)^3 = 200 \cdot \frac{1}{8} = 25 \text{ mg}$$
Worked Example 2.2 — Population Growth
A bacterial colony of 500 doubles every 3 hours. Write a function for the population $P(t)$ at time $t$ hours, and find the population after 12 hours.
Model $P(t) = 500 \cdot 2^{t/3}$.
Calculate $P(12) = 500 \cdot 2^{12/3} = 500 \cdot 2^4 = 500 \cdot 16 = 8000$ bacteria.
3. Discrete Functions: Sequences and Series
A sequence is an ordered list of numbers generated by a rule. The individual numbers are called terms, denoted $t_1, t_2, t_3, \ldots, t_n$. MCR3U focuses on two fundamental types.
Arithmetic Sequences
Arithmetic Sequence
Each term is obtained by adding a fixed value called the common difference $d$ to the previous term.
General term: $t_n = a + (n - 1)d$, where $a = t_1$ is the first term.
Sum of first $n$ terms: $S_n = \dfrac{n}{2}(2a + (n-1)d) = \dfrac{n}{2}(t_1 + t_n)$
Worked Example 3.1 — Arithmetic Sequence
The 4th term of an arithmetic sequence is 17 and the 9th term is 37. Find the common difference, the first term, and the sum of the first 20 terms.
Step 1 $t_9 - t_4 = 5d \Rightarrow 37 - 17 = 5d \Rightarrow d = 4$.
Step 2 $t_4 = a + 3d \Rightarrow 17 = a + 12 \Rightarrow a = 5$.
Step 3 $S_{20} = \dfrac{20}{2}(2(5) + 19(4)) = 10(10 + 76) = 10 \times 86 = 860$.
Geometric Sequences
Geometric Sequence
Each term is obtained by multiplying the previous term by a fixed value called the common ratio $r$.
General term: $t_n = ar^{n-1}$
Sum of first $n$ terms: $S_n = \dfrac{a(r^n - 1)}{r - 1}$, for $r \neq 1$
Sum of infinite geometric series (when $|r| < 1$): $S_\infty = \dfrac{a}{1 - r}$
Worked Example 3.2 — Geometric Series
A geometric series has first term 12 and common ratio $\dfrac{1}{3}$. Find the sum of the infinite series.
Check $|r| = \dfrac{1}{3} < 1$, so the infinite sum exists.
Apply $S_\infty = \dfrac{12}{1 - \frac{1}{3}} = \dfrac{12}{\frac{2}{3}} = 12 \times \dfrac{3}{2} = 18$.
Worked Example 3.3 — Connecting Sequences to Exponential Functions
Show that the $n$th term of a geometric sequence with first term $a$ and ratio $r$ is an exponential function of $n$.
$t_n = a \cdot r^{n-1} = \dfrac{a}{r} \cdot r^n$. This has the form $A \cdot b^n$ where $A = \dfrac{a}{r}$ and $b = r$, which is exactly an exponential function with base $r$.
4. Trigonometric Functions
The Unit Circle and Radian Measure
In MCR3U, trigonometric functions are extended beyond right triangles using the unit circle — a circle of radius 1 centred at the origin. For any angle $\theta$ measured from the positive $x$-axis:
- $\cos\theta$ is the $x$-coordinate of the terminal point
- $\sin\theta$ is the $y$-coordinate of the terminal point
- $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ (undefined when $\cos\theta = 0$)
Radian Measure
One radian is the angle subtended at the centre of a unit circle by an arc of length 1. Conversion: $$\pi \text{ rad} = 180°$$
So $1° = \dfrac{\pi}{180}$ rad and $1$ rad $= \dfrac{180°}{\pi} \approx 57.3°$.
| Degrees | Radians | $\sin$ | $\cos$ | $\tan$ |
|---|---|---|---|---|
| 0° | $0$ | $0$ | $1$ | $0$ |
| 30° | $\dfrac{\pi}{6}$ | $\dfrac{1}{2}$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{1}{\sqrt{3}}$ |
| 45° | $\dfrac{\pi}{4}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $1$ |
| 60° | $\dfrac{\pi}{3}$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{1}{2}$ | $\sqrt{3}$ |
| 90° | $\dfrac{\pi}{2}$ | $1$ | $0$ | undefined |
The CAST Rule
The CAST rule tells us which trig ratios are positive in each quadrant:
- Q1 (0° to 90°): All positive
- Q2 (90° to 180°): Sine positive
- Q3 (180° to 270°): Tangent positive
- Q4 (270° to 360°): Cosine positive
Transformations of Trig Functions
Sinusoidal Functions: $y = a\sin(k(x - d)) + c$
- $|a|$: amplitude (half the vertical range)
- $\dfrac{2\pi}{|k|}$: period
- $d$: phase shift (horizontal translation, right if $d > 0$)
- $c$: vertical shift (equation of the midline $y = c$)
Worked Example 4.1 — Analyzing a Sinusoidal Function
State the amplitude, period, phase shift, and midline of $y = 3\sin\!\left(2x - \dfrac{\pi}{3}\right) - 1$.
Step 1 Rewrite in standard form by factoring out $k = 2$: $$y = 3\sin\!\left(2\!\left(x - \frac{\pi}{6}\right)\right) - 1$$
- Amplitude $= |a| = 3$
- Period $= \dfrac{2\pi}{|k|} = \dfrac{2\pi}{2} = \pi$
- Phase shift $= \dfrac{\pi}{6}$ to the right
- Midline: $y = -1$; range is $[-4, 2]$
Worked Example 4.2 — Finding a Trig Equation from a Graph
A sinusoidal function has a maximum of 7, a minimum of 1, and a period of 8. Write an equation using the sine function.
Amplitude $a = \dfrac{7 - 1}{2} = 3$.
Midline $c = \dfrac{7 + 1}{2} = 4$.
Period $\dfrac{2\pi}{k} = 8 \Rightarrow k = \dfrac{\pi}{4}$.
One possible equation (assuming no phase shift): $y = 3\sin\!\left(\dfrac{\pi}{4}x\right) + 4$.
5. Practice Problems
Problem 1 — Domain
State the domain of $f(x) = \sqrt{5 - 2x}$.
Show Solution
Need $5 - 2x \geq 0 \Rightarrow x \leq \dfrac{5}{2}$. Domain: $\left(-\infty, \dfrac{5}{2}\right]$.
Problem 2 — Inverse Function
Find the inverse of $f(x) = \dfrac{2x + 1}{x - 3}$ and state any restrictions on the domain of $f^{-1}$.
Show Solution
Swap $x$ and $y$: $x = \dfrac{2y+1}{y-3}$. Solve: $x(y-3) = 2y+1$, so $xy - 3x = 2y + 1$, giving $y(x-2) = 3x+1$, thus $y = \dfrac{3x+1}{x-2}$.
$f^{-1}(x) = \dfrac{3x+1}{x-2}$, domain: $x \neq 2$.
Problem 3 — Exponential Decay
A 800 mg sample of iodine-131 has a half-life of 8 days. What amount remains after 24 days?
Show Solution
Number of half-lives: $24 \div 8 = 3$. Amount remaining: $800 \times \left(\dfrac{1}{2}\right)^3 = 800 \times \dfrac{1}{8} = 100$ mg.
Problem 4 — Arithmetic Series
Find the sum of all odd integers from 1 to 99 inclusive.
Show Solution
The sequence is $1, 3, 5, \ldots, 99$ with $a = 1$, $d = 2$. Number of terms: $n = \dfrac{99 - 1}{2} + 1 = 50$.
$S_{50} = \dfrac{50}{2}(1 + 99) = 25 \times 100 = 2500$.
Problem 5 — Geometric Series
Find the sum of the geometric series: $3 + 6 + 12 + \cdots + 384$.
Show Solution
$a = 3$, $r = 2$. Find $n$: $t_n = 3 \cdot 2^{n-1} = 384 \Rightarrow 2^{n-1} = 128 = 2^7 \Rightarrow n = 8$.
$S_8 = \dfrac{3(2^8 - 1)}{2 - 1} = 3 \times 255 = 765$.
Problem 6 — Exact Trig Values
Without a calculator, find the exact value of $\sin 150° + \cos 210° + \tan 315°$.
Show Solution
$\sin 150° = \sin(180° - 30°) = \sin 30° = \dfrac{1}{2}$ (Q2, sin positive)
$\cos 210° = \cos(180° + 30°) = -\cos 30° = -\dfrac{\sqrt{3}}{2}$ (Q3, cos negative)
$\tan 315° = \tan(360° - 45°) = -\tan 45° = -1$ (Q4, tan negative)
Sum $= \dfrac{1}{2} - \dfrac{\sqrt{3}}{2} - 1 = \dfrac{1 - \sqrt{3}}{2} - 1 = \dfrac{-1 - \sqrt{3}}{2}$.
Problem 7 — Transformations
The function $y = f(x)$ has domain $[-2, 5]$ and range $[0, 8]$. State the domain and range of $y = 2f(3x) - 1$.
Show Solution
Horizontal compression by factor $\frac{1}{3}$: domain becomes $\left[\dfrac{-2}{3}, \dfrac{5}{3}\right]$.
Vertical stretch by 2 and shift down 1: range becomes $[2(0)-1, 2(8)-1] = [-1, 15]$.
Problem 8 — Sinusoidal Application
The height of a Ferris wheel seat (in metres) above the ground is modelled by $h(t) = 12\sin\!\left(\dfrac{\pi}{15}(t - 7.5)\right) + 14$, where $t$ is time in seconds. Find: (a) the radius of the Ferris wheel, (b) its period, (c) the minimum height.
Show Solution
(a) Amplitude $= 12$ m, so the radius is 12 m.
(b) Period $= \dfrac{2\pi}{\pi/15} = 30$ seconds.
(c) Minimum height $= 14 - 12 = $ 2 m (the board at the bottom).