MCR3U · Unit 1 test review

Characteristics of Functions: Test Review

Everything for the Unit 1 test on one page: the key facts from all 12 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Relations vs. Functions

A relation is a function iff every input has exactly one output (no repeated $x$ with different $y$ ).
Vertical line test: if a vertical line crosses the graph more than once, it is not a function.
$f (x)$ names the output of $f$ at input $x$ ; it is not $f \times x$ .
To evaluate $f (a)$ , replace every $x$ in the rule with $a$ , then simplify.
Independent variable = input = $x$ (domain); dependent variable = output = $f (x)$ or $y$ (range).
To solve $f (x) = k$ , set the rule equal to $k$ and solve for $x$ ; a many-to-one function can give several solutions.

Domain and Range

Domain = every valid input ( $x$ ); Range = every possible output ( $y$ ).
Radical restriction: the radicand of an even root must be $\geq 0$ .
Denominator restriction: a denominator can never equal $0$ .
Interval notation: square bracket $[]$ = included endpoint, round bracket $()$ = excluded endpoint; $\pm \infty$ always gets a round bracket.
On a graph, domain is read along the $x$ -axis and range along the $y$ -axis.
For $y = a (x - h)^{2} + k$ , the range is $y \geq k$ when $a > 0$ and $y \leq k$ when $a < 0$ .

Parent Functions

$y = x$ : domain and range are all real numbers; it's the only parent function that's a straight line.
$y = x^{2}$ : domain all reals, range $y \geq 0$ ; vertex (turning point) at the origin.
$y = x$ : domain $x \geq 0$ , range $y \geq 0$ ; starts at the origin.
$y = \frac{1}{x}$ : domain and range both exclude $0$ ; has asymptotes along both axes.
$y = ∣ x ∣$ : domain all reals, range $y \geq 0$ ; sharp corner (not smooth) at the origin.
$y = x$ is the right half of $y = x^{2}$ reflected in $y = x$ : key points $(0, 0), (1, 1), (4, 2), (9, 3)$ .

Transformations of Functions

$y = a f (k (x - d)) + c$ : $a$ = vertical stretch/reflection, $k$ = horizontal stretch/reflection, $d$ = horizontal shift, $c$ = vertical shift.
Mapping rule: $(x, y) \to (\frac{x}{k} + d, a y + c)$ .
Negative $a$ reflects in the $x$ -axis; negative $k$ reflects in the $y$ -axis.
Horizontal stretch/compression factor is $\frac{1}{k}$ (the reciprocal of $k$ ), not $k$ itself.
Apply stretches and reflections before translations — that's exactly the order built into the mapping rule.
Factor $k$ out of the bracket before reading $d$ : $f (2 x - 6) = f (2 (x - 3))$ gives $d = 3$ .

Inverse Functions

To find $f^{- 1} (x)$ algebraically: replace $f (x)$ with $y$ , swap $x$ and $y$ , then solve for $y$ .
$f$ and $f^{- 1}$ are reflections of each other in the line $y = x$ ; if $(a, b)$ is on $f$ , then $(b, a)$ is on $f^{- 1}$ .
$f^{- 1} (x)$ is notation for the inverse function, not $\frac{1}{f ( x )}$ .
The inverse of $f$ is a function only if $f$ passes the horizontal line test (i.e., $f$ is one-to-one).
Restricting the domain of a many-to-one function (like $x^{2}$ ) can make its inverse a function.
Domain of $f^{- 1}$ = range of $f$ , and range of $f^{- 1}$ = domain of $f$ ; verify with $f (f^{- 1} (x)) = x$ .

Forms of a Quadratic Relation

Standard form $y = a x^{2} + b x + c$ : $c$ is the $y$ -intercept, sign of $a$ gives opening direction.
Factored form $y = a (x - r_{1}) (x - r_{2})$ : $r_{1}, r_{2}$ are the $x$ -intercepts.
Vertex form $y = a (x - h)^{2} + k$ : $(h, k)$ is the vertex (max/min point).
Axis of symmetry from factored form: $x = \frac{r _{1} + r _{2}}{2}$ ; from vertex form: $x = h$ ; from standard form: $x = \frac{- b}{2 a}$ .
All three forms describe the same parabola — converting between them never changes the graph, only how it's written.

Completing the Square

To complete the square on $x^{2} + b x$ : add and subtract $(\frac{b}{2})^{2}$ .
When $a \neq = 1$ , factor $a$ out of the $x^{2}$ and $x$ terms first, before completing the square.
After completing the square, the constant subtracted outside the bracket must be multiplied by $a$ .
Vertex form result: $y = a (x - h)^{2} + k$ , vertex $(h, k)$ .
If $a > 0$ , $k$ is a minimum value; if $a < 0$ , $k$ is a maximum value.

Solving Quadratic Equations

Zero product property: if $A \cdot B = 0$ then $A = 0$ or $B = 0$ — only valid when one side of the equation is exactly $0$ .
Quadratic formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ , valid for any $a x^{2} + b x + c = 0$ .
Discriminant: $D = b^{2} - 4 a c$ .
$D > 0$ : two real roots; $D = 0$ : one repeated real root; $D < 0$ : no real roots.
Before factoring or using the formula, always set the equation equal to $0$ first.

Quadratic Applications

Projectile height model: $h (t) = - 4.9 t^{2} + v_{0} t + h_{0}$ (metres, seconds).
Revenue model: $Revenue = price \times quantity$ .
The vertex of a quadratic model gives the maximum (if $a < 0$ ) or minimum (if $a > 0$ ) value of the situation.
The $t$ - or $x$ -coordinate of the vertex tells you when or at what input the max/min happens; the $y$ -coordinate tells you the max/min value itself.
Always interpret the vertex back in context — include correct units in the final answer.

Radicals

Simplify an entire radical by factoring out the largest perfect square: $72 = 62$ .
Only like radicals combine: $a n + b n = (a + b) n$ .
$a \cdot b = ab$ ; multiply coefficients and radicands separately.
Rationalize a single-term radical denominator by multiplying top and bottom by that radical.
Rationalize a binomial radical denominator by multiplying top and bottom by its conjugate: $(A + B) (A - B) = A^{2} - B^{2}$ .

Rational Expressions

Always factor completely before cancelling, multiplying, dividing, or finding a common denominator.
Restrictions come from every denominator in the original (un-cancelled) expression.
Dividing rational expressions: multiply by the reciprocal of the second fraction.
Adding/subtracting rational expressions: rewrite over a common denominator, then combine numerators only.
A cancelled factor still creates a restriction — cancelling changes the simplified form, not the domain.

Equivalent Algebraic Expressions

Two expressions are equivalent if and only if they produce the same output for every value of the variable.
To prove equivalence, fully expand and simplify both expressions, then compare term by term.
A single matching test value is not proof of equivalence — it can only disprove it (if they disagree there).
$(A + B)^{2} = A^{2} + 2 A B + B^{2}$ — the middle term is easy to forget.
$(A - B) (A + B) = A^{2} - B^{2}$ (difference of squares) — this is different from $(A - B)^{2}$ .

Where marks get lost

Reading $f (x)$ as " $f$ times $x$ " instead of "the output of $f$ at $x$ ." (Relations vs. Functions)
Assuming any equation or graph is automatically a function without checking (e.g. treating $x = y^{2}$ as a function of $x$ ). (Relations vs. Functions)
Forgetting to keep brackets when substituting an expression like $a + 1$ , leading to sign errors (writing $- 3 a + 1$ instead of correctly expanding $- 3 (a + 1) = - 3 a - 3$ ). (Relations vs. Functions)
Declaring a table "not a function" just because a $y$ -value repeats — repeated outputs are allowed; only repeated inputs with different outputs are not. (Relations vs. Functions)
Assuming $f (x + 2) = f (x) + 2$ . For $f (x) = x^{2}$ , $f (x + 2) = (x + 2)^{2} = x^{2} + 4 x + 4$ , which is not $x^{2} + 2$ — substitute into the rule, never "add to the output." (Relations vs. Functions)
Setting the whole function $\geq 0$ instead of just the radicand when finding the domain of a square-root function. (Domain and Range)
Forgetting the denominator restriction when the denominator is a more complex expression like $x^{2} - 9$ (should exclude both $x = 3$ and $x = - 3$ ). (Domain and Range)
Mixing up which axis gives domain and which gives range when reading a graph. (Domain and Range)
Using a square bracket next to $\infty$ or $- \infty$ . (Domain and Range)
Giving the algebraic domain for a real-world model — for example "all real numbers" for a height model, when time must be between $0$ and the landing time. (Domain and Range)
Thinking $y = x^{2}$ and $y = ∣ x ∣$ are the same graph because they share domain and range — the shapes differ (curve vs. sharp corner). (Parent Functions)
Forgetting that $y = x$ only exists for $x \geq 0$ , and trying to evaluate it at a negative $x$ . (Parent Functions)
Believing $y = 1/ x$ passes through the origin — it never does; the origin is exactly where it's undefined. (Parent Functions)
Confusing the vertex of $y = x^{2}$ (a smooth minimum) with the corner of $y = ∣ x ∣$ (a sharp minimum) when sketching by hand. (Parent Functions)
Evaluating absolute value by "dropping the signs" of each number: $∣2 - 7∣$ is $∣ - 5∣ = 5$ , not $2 + 7 = 9$ . Simplify inside the bars first. (Parent Functions)
Using $k$ directly as the horizontal stretch factor instead of $\frac{1}{k}$ (e.g. treating $k = 2$ as "stretch by 2" when it's actually a compression by $\frac{1}{2}$ ). (Transformations of Functions)
Forgetting that $d$ must be found by writing the bracket as $(x - d)$ — misreading $f (x + 3)$ as $d = 3$ instead of $d = - 3$ . (Transformations of Functions)
Applying the translation before the stretch/reflection, which gives the wrong image points. (Transformations of Functions)
Forgetting to reflect when $a$ or $k$ is negative — only scaling by the absolute value and skipping the flip. (Transformations of Functions)
Reading $d$ before factoring out $k$ : $f (2 x + 6)$ is $f (2 (x + 3))$ , a shift of 3 left, not 6 left. (Transformations of Functions)
Confusing $f^{- 1} (x)$ with $\frac{1}{f ( x )}$ — they are unrelated. (Inverse Functions)
Forgetting to swap $x$ and $y$ before solving — just solving the original equation for $x$ instead. (Inverse Functions)
Taking only the positive square root without checking whether the original domain restriction actually calls for it (sometimes the negative root is the correct branch). (Inverse Functions)
Assuming every function has an inverse function without first checking the horizontal line test. (Inverse Functions)
Giving the inverse's domain from its own formula alone. For $f (x) = x^{2} + 4, x \geq 0$ , the inverse $x - 4$ has domain $x \geq 4$ — which is the range of $f$ ; always swap the original domain and range. (Inverse Functions)
Reading $k$ from vertex form as the $y$ -intercept. In $y = 3 (x - 2)^{2} + 5$ the $y$ -intercept is $3 (4) + 5 = 17$ , not $5$ ; the constant equals the $y$ -intercept only in standard form. (Forms of a Quadratic Relation)
Forgetting to distribute a negative sign when expanding factored form with a leading negative, like $- (x + 2) (x - 6)$ . (Forms of a Quadratic Relation)
Reading the zeros with the wrong sign: $y = 5 (x + 1) (x - 4)$ has zeros $x = - 1$ and $x = 4$ , because each zero is the value that makes its bracket equal zero. (Forms of a Quadratic Relation)
Mixing up the axis-of-symmetry formulas for the three forms (using $\frac{- b}{2 a}$ on factored form instead of $\frac{r _{1} + r _{2}}{2}$ ). (Forms of a Quadratic Relation)
Writing $y = (x + 1) (x - 5)$ for "zeros $- 1$ and $5$ through $(1, - 16)$ " and forgetting to solve for $a$ . The given point is what fixes $a$ , here $a = 2$ . (Forms of a Quadratic Relation)
Forgetting to factor $a$ out of the $x$ -term before completing the square when $a \neq = 1$ . (Completing the Square)
Subtracting the completed-square number directly instead of multiplying it by $a$ first when distributing back out. (Completing the Square)
Sign errors when halving a negative $b$ (e.g. halving $- 4$ should give $- 2$ , not $2$ ). (Completing the Square)
Reporting $k$ as the maximum when $a > 0$ (it's actually the minimum in that case). (Completing the Square)
Reading the vertex of $y = (x + 3)^{2} - 2$ as $(3, - 2)$ . The bracket $(x + 3)$ means $h = - 3$ , so the vertex is $(- 3, - 2)$ . (Completing the Square)
Trying to use the zero product property when the equation isn't set equal to $0$ first (e.g. factoring $x^{2} - 5 x = 6$ without moving the 6 over). (Solving Quadratic Equations)
Sign errors substituting a negative $b$ into $- b$ in the quadratic formula (forgetting $- (- 12) = 12$ ). (Solving Quadratic Equations)
Forgetting the $\pm$ and reporting only one root when $D > 0$ . (Solving Quadratic Equations)
Concluding "no solution" when $D < 0$ without clarifying it means no real solution (not that the equation is invalid). (Solving Quadratic Equations)
Dividing both sides of $x^{2} = 5 x$ by $x$ to get $x = 5$ , which loses the root $x = 0$ . Rearrange to $x^{2} - 5 x = 0$ and factor: $x (x - 5) = 0$ . (Solving Quadratic Equations)
Forgetting the negative sign on the $- 4.9 t^{2}$ term in projectile models — gravity must pull the object down, so $a$ is always negative. (Quadratic Applications)
Reporting the vertex's $x$ -coordinate as the answer when the question actually asks for the maximum value (the $y$ -coordinate), or vice versa. (Quadratic Applications)
In fencing/area problems, using the wrong total when one side doesn't need fencing (forgetting a wall or existing side reduces the amount of fencing needed). (Quadratic Applications)
In revenue problems, only adjusting the price or only adjusting the quantity, instead of writing both as expressions in the same variable. (Quadratic Applications)
Keeping an impossible root. For $h (t) = - 4.9 t^{2} + 14.7 t + 2 = 0$ the formula gives $t \approx 3.13$ and $t \approx - 0.13$ ; the negative time is before the launch and must be rejected. (Quadratic Applications)
Trying to add radicals that aren't alike without simplifying first (e.g. leaving $50$ and $2$ unsimplified and concluding they can't combine). (Radicals)
Multiplying by the same denominator again instead of its conjugate when rationalizing a binomial denominator. (Radicals)
Forgetting to distribute across both terms of the numerator when multiplying by a conjugate. (Radicals)
Picking a factor that isn't the largest perfect square (e.g. writing $72 = 218$ and stopping, instead of continuing to $62$ ). (Radicals)
Splitting a root over a sum: writing $9 + 16 = 9 + 16 = 7$ . The product rule has no sum version; $25 = 5$ . (Radicals)
Cancelling terms that are added or subtracted instead of only cancelling common factors (e.g. wrongly cancelling the $x$ 's in $\frac{x + 3}{x}$ ). (Rational Expressions)
Stating restrictions only from the simplified expression's denominator, missing restrictions from factors that cancelled. (Rational Expressions)
Forgetting to flip the second fraction (multiply by the reciprocal) when dividing rational expressions. (Rational Expressions)
Adding numerators without first getting a common denominator. (Rational Expressions)
Forgetting the restriction created by the divisor's numerator when dividing (it can't equal zero either). (Rational Expressions)
Expanding $(x - 4)^{2}$ as $x^{2} - 16$ — confusing a squared binomial with a difference of squares; the correct expansion always has a middle term. (Equivalent Algebraic Expressions)
Concluding two expressions are equivalent after checking only one value of $x$ . (Equivalent Algebraic Expressions)
Forgetting to distribute a negative sign across every term when expanding something like $- 2 (x^{2} - 3 x + 1)$ . (Equivalent Algebraic Expressions)
Stopping after expanding only one of the two expressions and assuming a visual "similarity" is enough. (Equivalent Algebraic Expressions)
Multiplying the coefficient in before squaring: writing $2 (x - 3)^{2}$ as $(2 x - 6)^{2} = 4 x^{2} - 24 x + 36$ . Square first: $2 (x^{2} - 6 x + 9) = 2 x^{2} - 12 x + 18$ . (Equivalent Algebraic Expressions)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Relations vs. Functions
1
Which set of ordered pairs is NOT a function?
Relations vs. Functions
2
A mapping diagram shows the input 6 with two arrows, one to 1 and one to 8. Is the relation a function? Answer yes or no.
Domain and Range
3
State the domain of $f (x) = 2 x + 6$ using an inequality.
Domain and Range
4
A rock dropped from a cliff has height $h (t) = 80 - 5 t^{2}$ metres. What is the largest value in the domain of this model (the time it lands), in seconds?
s
Parent Functions
5
Match each parent function to its domain and range.
y = x
y = x²
y = √x
y = 1/x
Parent Functions
6
The point $(3, 9)$ lies on $y = x^{2}$ . Which point on $y = x$ corresponds to it?
Transformations of Functions
7
For $g (x) = 4 f (x) - 7$ , which best describes the transformation?
Transformations of Functions
8
State the range of $g (x) = - 3 x + 2 + 7$ as an inequality.
Inverse Functions
9
Find $f^{- 1} (x)$ for $f (x) = 5 x + 10$ .
Inverse Functions
10
Match each function with its inverse.
1. A. $\frac{x}{7}$
2. B. $\frac{x + 1}{7}$
3. C. $7 (x - 1)$
4. D. $x - 7$
$f (x) = x + 7$
$f (x) = 7 x$
$f (x) = 7 x - 1$
$f (x) = \frac{x}{7} + 1$
Forms of a Quadratic Relation
11
For $y = 5 (x + 1) (x - 4)$ , what are the $x$ -intercepts?
Forms of a Quadratic Relation
12
Match each equation with the feature you can read from it directly.
1. A. $y$ -intercept $1$
2. B.opens down, vertex $(- 3, 0)$
3. C.vertex $(3, 1)$
4. D.zeros $3$ and $- 1$
$y = (x - 3) (x + 1)$
$y = (x - 3)^{2} + 1$
$y = x^{2} - 3 x + 1$
$y = - (x + 3)^{2}$
Completing the Square
13
Complete the square: $y = x^{2} - 10 x + 30$ . State the vertex.
Completing the Square
14
Which is the completed-square form of $y = x^{2} + 5 x + 1$ ?
Solving Quadratic Equations
15
Solve by factoring: $x^{2} - 7 x + 12 = 0$ .
Solving Quadratic Equations
16
Match each equation with its discriminant.
1. A. $- 16$
2. B. $0$
3. C. $16$
4. D. $5$
$x^{2} + 4 = 0$
$x^{2} - 4 = 0$
$x^{2} - 4 x + 4 = 0$
$x^{2} + x - 1 = 0$
Quadratic Applications
17
A ball's height is $h (t) = - 4.9 t^{2} + 24.5 t + 2$ . To the nearest 0.1 s, at what time is the ball at its maximum height?
s
Quadratic Applications
18
For $h (t) = - 4.9 t^{2} + 19.6 t$ , match each question with the correct answer.
How high is the peak?
When is the peak?
When does it land?
How high is it at launch?
Radicals
19
Simplify $98$ .
Radicals
20
Expand and simplify $(3 + 2) (1 + 2)$ into the form $a + b 2$ . What is $a$ ?
Rational Expressions
21
Simplify $\frac{x ^{2} - 16}{x + 4}$ and state the restriction.
Rational Expressions
22
Simplify $\frac{5}{x + 1} - \frac{2}{x - 3}$ .
Equivalent Algebraic Expressions
23
Which expression is equivalent to $(x + 5)^{2}$ ?
Equivalent Algebraic Expressions
24
Which expression is NOT equivalent to $2 x^{2} - 5 x - 12$ ?

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

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Characteristics of Functions: Test Review

Key facts

Where marks get lost

Practice test

Endless drill

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