VertexMath · MHF4U · Free worksheet

Functions Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Even: $f(-x)=f(x)$ , symmetric in the $y$ -axis. Odd: $f(-x)=-f(x)$ , symmetric about the origin. Test with $f(-x)$ , not by looking at exponents.
Transformations: in $y=af(k(x-d))+c$ , factor $k$ out first. $y=f(2x+8)=f(2(x+4))$ shifts the graph 4 units left, not 8.
Inverse: swap $x$ and $y$ , then solve for $y$ . The domain of $f^{-1}$ is the range of $f$ .
Rate of change: the average rate on $[a,b]$ is $\dfrac{f(b)-f(a)}{b-a}$ . The instantaneous rate is the value $\dfrac{f(a+h)-f(a)}{h}$ approaches as $h\to0$ .
In $f(x+h)-f(x)$ , the minus sign applies to every term of $f(x)$ .

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Domain · Transformations · Piecewise · Inverses · Rates of Change

K1 Domain First

Let $f(x)=\dfrac{x^2-x-6}{x-3}$ .

a)
State the domain of $f$ in interval notation.
b)
Simplify $f(x)$ . Then evaluate $f(1)$ and $f(3)$ , if they exist.
c)
State the domain of $g(x)=\dfrac{\sqrt{8-2x}}{x+1}$ in interval notation.

Stuck? Peek at a hint

Find every restriction from the original rule, before cancelling anything: no zero denominators, nothing negative under a square root.

K2 Where Does the Point Go?

The graph of $y=f(x)$ is shown. Each grid square is 1 unit. Find the image of the given point on the graph of each new function.

a)
Point $B$ , for $y=f(x+3)-2$ .
b)
Point $C$ , for $y=-2f(-x)$ .
c)
Point $D$ , for $y=3f(2x+12)-1$ .

Stuck? Peek at a hint

Set the expression inside the brackets equal to the point's original $x$ -coordinate and solve for $x$ . The operations outside the brackets act on $y$ .

K3 Piecewise Functions

Let
$f(x)=\begin{cases} x^2+2, & x<1 \\ 1-x, & x\ge 1 \end{cases}$

a)
Evaluate $f(-2)$ , $f(1)$ and $f(4)$ .
b)
State the domain and range of $f$ in interval notation.
c)
Determine whether $f$ is continuous at $x=1$ . If it is not, find the size of the jump.

Stuck? Peek at a hint

Before substituting, check which piece's condition the input satisfies.

K4 Inverses

a)
The mapping diagram shows a function $f$ . Evaluate $f^{-1}(-3)$ and $f^{-1}(7)$ .
b)
Determine $g^{-1}(x)$ for $g(x)=2(x-1)^3+5$ .
c)
Determine $h^{-1}(x)$ for $h(x)=3-\sqrt{x+2}$ , and state its domain and range.

Stuck? Peek at a hint

Swap $x$ and $y$ , then undo the operations on $y$ starting with the last one applied. For (c), find the range of $h$ first.

K5 Rates of Change

Let $f(x)=x^2-3x+4$ .

a)
Find the average rate of change of $f$ on $[1,4]$ .
b)
Simplify the difference quotient $\dfrac{f(x+h)-f(x)}{h}$ , for $h\ne0$ .
c)
Use (b) to find the instantaneous rate of change of $f$ at $x=2$ .

Stuck? Peek at a hint

In (b), replace every $x$ in the rule with $(x+h)$ , expand, then subtract all of $f(x)$ .

Real World (A)

Rates in Context · Inverses in Context · Piecewise Pricing

A1 Battery Drain

Priya records her phone's battery level $B$ , in %, every 2 hours during a long day. $t$ is the time in hours.

$t$	$0$	$2$	$4$	$6$	$8$
$B$	$100$	$84$	$64$	$42$	$22$

a)
Find the average rate of change of $B$ from $t=0$ to $t=4$ , with units. What does it mean for the phone?
b)
Over which 2-hour interval did the battery drain fastest? Give the rate.
c)
Use the rate from $t=6$ to $t=8$ to predict the battery level at $t=11$ . Does the prediction make sense?

Stuck? Peek at a hint

Divide the change in battery level by the change in time.

A2 Skid Marks

On a wet road, the braking distance of a car, in metres, at a speed of $v$ km/h is modelled by $d(v)=\dfrac{v^2}{200}$ .

a)
Determine $d^{-1}(x)$ for $v\ge0$ . What do its input and its output represent?
b)
Police measure skid marks 72 m long in a 100 km/h zone. How fast was the car going? Was it speeding?
c)
Solving $\dfrac{v^2}{200}=72$ gives $v=\pm120$ . Why does $d$ need the restriction $v\ge0$ to have an inverse function, and why is $v=-120$ rejected?

Stuck? Peek at a hint

Write $x=\dfrac{v^2}{200}$ and solve for $v$ .

A3 Tiered Electricity

An Ontario utility charges $0.10 per kWh for the first 600 kWh used in a month, and $0.12 per kWh for every kWh above 600. Let $C(E)$ be the monthly cost, in dollars, for $E$ kWh.

a)
Write $C(E)$ as a piecewise function. Find the cost of using 900 kWh.
b)
A monthly bill is $84. How much electricity was used?
c)
Plan B charges $0.10 per kWh if $E\le600$ , but $0.12 per kWh on all usage if $E>600$ . A customer on plan B says their bill was $66. Show that this is impossible.

Stuck? Peek at a hint

A bill is an output of $C$ : decide which piece can produce it before solving.

Think It Through (T)

Counterexamples · Error Analysis

T1 Even, Odd, or Neither?

A student claims: "If the only power of $x$ in a function's rule is even, the function is even."

a)
Show that $p(x)=(x-3)^2$ is a counterexample. Is $p$ even, odd, or neither?
b)
Determine whether $q(x)=\dfrac{x^2+1}{x}$ is even, odd, or neither, using $q(-x)$ .
c)
Explain why the only function with domain $\mathbb{R}$ that is both even and odd is $f(x)=0$ .

Stuck? Peek at a hint

Compare $p(-3)$ with $p(3)$ .

T2 Spot the Flaw

A student describes how to get $y=-2\sqrt{4-x}+1$ from $y=\sqrt{x}$ :
"Reflect in the $x$ -axis, stretch vertically by a factor of 2, translate 4 units left, then 1 unit up."

a)
Apply the student's steps to the point $(9,3)$ on $y=\sqrt{x}$ . Show that the image is not on the new graph.
b)
Write the equation in the form $y=af(k(x-d))+c$ and give the correct list of transformations. Find the correct image of $(9,3)$ .
c)
State the domain and range of $y=-2\sqrt{4-x}+1$ in interval notation.

Stuck? Peek at a hint

Factor $-1$ out of $4-x$ before reading off the horizontal transformations.

Boss Level

Optional challenge

Nested Absolute Values

Consider the equation $\left||x-2|-3\right|=k$ .

a)
Solve the equation for $k=1$ .
b)
Find every value of $k$ for which the equation has exactly three solutions.
c)
For $0<k<3$ the equation has four solutions. Find their sum.

Stuck? Peek at a hint

Let $u=|x-2|-3$ and solve $|u|=k$ first; a sketch of $y=\left||x-2|-3\right|$ helps with (b) and (c).

Answer key

Final answers only. Try first, then check.

Core Skills

a) $(-\infty,3)\cup(3,\infty)$
b) $f(x)=x+2$ , $x\ne3$ ; $f(1)=3$ ; $f(3)$ does not exist
c) $(-\infty,-1)\cup(-1,4]$

a) $(-5,3)$
b) $(-2,6)$
c) $(-3,2)$

a) $f(-2)=6$ , $f(1)=0$ , $f(4)=-3$
b) $D=(-\infty,\infty)$ , $R=(-\infty,0]\cup[2,\infty)$
c) Not continuous: the left piece approaches $3$ but $f(1)=0$ , a jump of $3$ units

a) $f^{-1}(-3)=3$ ; $f^{-1}(7)=-2$
b) $g^{-1}(x)=1+\sqrt[3]{\dfrac{x-5}{2}}$
c) $h^{-1}(x)=(3-x)^2-2$ ; $D=(-\infty,3]$ , $R=[-2,\infty)$

a) $2$
b) $2x+h-3$
c) $1$

Real World

a) $-9$ percentage points per hour: on average, the battery lost 9 % of its charge each hour
b) From $t=4$ to $t=6$ : $-11$ percentage points per hour
c) $22-3(10)=-8$ %, which is impossible: at 10 points per hour the battery reaches 0 % at about $t=10.2$ h, so the model stops there

a) $d^{-1}(x)=\sqrt{200x}$ ; input: a braking distance in m; output: the speed in km/h
b) $120$ km/h: yes, 20 km/h over the limit
c) On all real numbers $d(-120)=d(120)=72$ , so $d$ fails the horizontal line test. A speed cannot be negative, so the model only applies for $v\ge0$ , where $d$ is one-to-one.

a) $C(E)=0.10E$ for $0\le E\le600$ ; $C(E)=60+0.12(E-600)$ for $E>600$ ; $96
b) $800$ kWh
c) $0.10E=66$ gives $E=660>600$ and $0.12E=66$ gives $E=550\le600$ : neither piece works. Plan B jumps from $60 at 600 kWh to above $72, so no bill lies in $(60,72]$ .

Think It Through

a) $p(-3)=36$ but $p(3)=0$ , so $p(-x)\ne p(x)$ ; $p$ is neither (also $p(-3)\ne-p(3)$ )
b) $q(-x)=\dfrac{x^2+1}{-x}=-q(x)$ , so $q$ is odd
c) Both conditions give $f(-x)=f(x)$ and $f(-x)=-f(x)$ , so $f(x)=-f(x)$ , $2f(x)=0$ and $f(x)=0$ for every $x$ .

a) The steps give $(5,-5)$ , but at $x=5$ the rule needs $\sqrt{-1}$ , which is undefined
b) $y=-2\sqrt{-(x-4)}+1$ : vertical stretch by 2, reflection in the $x$ -axis, reflection in the $y$ -axis, translation 4 right and 1 up; $(-5,-5)$
c) $D=(-\infty,4]$ , $R=(-\infty,1]$

Boss Level

Boss

a) $x=-2,\ 0,\ 4,\ 6$
b) $k=3$ only (solutions $-4,\ 2,\ 8$ )
c) $8$

Functions Review

Core Skills (K)

K1 Domain First

K2 Where Does the Point Go?

K3 Piecewise Functions

K4 Inverses

K5 Rates of Change

Real World (A)

A1 Battery Drain

A2 Skid Marks

A3 Tiered Electricity

Think It Through (T)

T1 Even, Odd, or Neither?

T2 Spot the Flaw

Boss Level

Nested Absolute Values

Answer key

Core Skills

Real World

Think It Through

Boss Level

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