VertexMath · MCR3U · Free worksheet

Introduction to Functions Review

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

Core path ~55 min Full sheet ~1 h 50 min

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

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

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Core Skills (K)

Relations · Notation · Parent Functions · Domain & Range · Inverses

K1 Function or Not?

For each relation, state its domain and range, then decide whether it is a function. If it is not, name one input that has two outputs.

a)
The mapping diagram.
b)
The table of values.

$x$ $3$ $5$ $3$ $7$
$y$ $1$ $2$ $4$ $2$
c)
The graph. Each grid square is 1 unit.

$x$	$3$	$5$	$3$	$7$
$y$	$1$	$2$	$4$	$2$

Stuck? Peek at a hint

Look for one input paired with two different outputs. For the graph, slide a vertical line across it.

K2 Function Notation

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

a)
Evaluate $f(-2)$ .
b)
Simplify $f(x+1)$ .
c)
Simplify $f(x)+1$ .

Stuck? Peek at a hint

Put brackets around whatever replaces $x$ : in (b), every $x$ in the rule becomes $(x+1)$ .

K3 Parent Functions

Graphs A to D show $y=x^2$ , $y=\sqrt{x}$ , $y=\dfrac{1}{x}$ and $y=|x|$ , in some order. Each grid square is 1 unit.

Graph A

Graph B

Graph C

Graph D

a)
Match each graph to its equation.
b)
State the domain and range of each function in interval notation.
c)
$y=x^2$ and $y=|x|$ have the same domain and range. Evaluate both at $x=3$ and at $x=\frac{1}{2}$ .

Stuck? Peek at a hint

Start with the features: which graph has two separate pieces, and which one starts at the origin and only goes right?

K4 Domain & Range

State the domain and range of each function in interval notation.

a)
$f(x)=\sqrt{10-2x}$
b)
$g(x)=\dfrac{3}{x+4}-1$
c)
The function graphed below. Each grid square is 1 unit.

Stuck? Peek at a hint

For a square root, set the expression under the root $\ge 0$ . Dividing an inequality by a negative number flips its sign.

K5 Inverses

a)
Determine $f^{-1}(x)$ for $f(x)=\dfrac{2x+1}{5}$ .
b)
Evaluate $f^{-1}(3)$ and $\dfrac{1}{f(3)}$ .
c)
For $q(x)=\sqrt{x+3}$ , determine $q^{-1}(x)$ and state its domain and range.

Stuck? Peek at a hint

Write $y=f(x)$ , swap $x$ and $y$ , then undo the operations in reverse order.

Real World (A)

Domain & Range in Context · Inverses in Context

A1 Balcony Toss

A ball is tossed upward from a balcony. Its height above the ground, in metres, after $t$ seconds is $h(t)=-5t^2+15t+20$ .

a)
Find when the ball hits the ground. Then state the domain and range of $h$ in this context.
b)
Solve $h(t)=20$ . What does each solution mean for the ball?
c)
The formula also gives $h(-1)=0$ . What, if anything, does this tell you about the ball?

Stuck? Peek at a hint

The ball's flight starts at $t=0$ and ends when $h(t)=0$ .

A2 Celsius to Fahrenheit

The function $F(c)=1.8c+32$ converts a temperature of $c$ °C into degrees Fahrenheit.

a)
Determine $F^{-1}(x)$ . What do its input and its output represent?
b)
A weather app shows 86 °F for Orlando. Use $F^{-1}$ to convert it to Celsius.
c)
Nothing can be colder than absolute zero, $-273.15$ °C. State the realistic domain and range of $F$ . Then evaluate $F^{-1}(-500)$ and say what the result means.

Stuck? Peek at a hint

Write $x=1.8c+32$ and solve for $c$ .

A3 Parking Cap

A downtown parking lot charges $4 per hour, charged for the exact time parked, up to a daily maximum of $20. Let $C(h)$ be the cost, in dollars, of parking for $h$ hours, where $0\le h\le 24$ .

a)
Write $C(h)$ as a piecewise function and state its range.
b)
You paid $14. How long did you park?
c)
Your friend paid $20. Can you tell how long they parked? Explain using the inverse of $C$ .

Stuck? Peek at a hint

In (b), $14 is a cost, so it is an output of $C$ , not an input.

Think It Through (T)

Error Analysis · Building Functions

T1 Spot the Flaw

A student finds the inverse of $f(x)=(x+2)^2-1$ :
"Swap $x$ and $y$ : $x=(y+2)^2-1$ , so $x+1=(y+2)^2$ and $y+2=\sqrt{x+1}$ . Therefore $f^{-1}(x)=\sqrt{x+1}-2$ , and $f^{-1}$ is a function."

a)
Find two different inputs that $f$ sends to $3$ . What does this mean for the inverse at $3$ ?
b)
Identify the step where the student lost a solution, and correct it.
c)
Restrict the domain of $f$ so that the student's formula is correct. Then give the other restriction and its inverse formula.

Stuck? Peek at a hint

For (a), solve $(x+2)^2-1=3$ .

T2 Build It Backwards

Write the equation of a function with each property. Many answers are possible.

a)
Domain $\{x\in\mathbb{R}\mid x\ne 3\}$ and range $\{y\in\mathbb{R}\mid y\ne -2\}$ .
b)
Domain $(-\infty,4]$ and range $[1,\infty)$ .
c)
It is its own inverse, and it is not $y=x$ . Show that $f(f(x))=x$ .

Stuck? Peek at a hint

Start from a parent function whose domain or range has the right shape, then shift or reflect it.

Boss Level

Optional challenge

A Function in Disguise

A function $f$ satisfies $f(2x+1)=4x^2+2x+3$ for every real number $x$ .

a)
Find $f(3)$ without finding a formula for $f$ .
b)
Find a formula for $f(x)$ .
c)
State the range of $f$ . For which value of $k$ does $f(x)=k$ have exactly one solution?

Stuck? Peek at a hint

For (a), choose $x$ so that $2x+1=3$ . For (b), let $u=2x+1$ and write $x$ in terms of $u$ .

Answer key

Final answers only. Try first, then check.

Core Skills

a) $D=\{-2,-1,1,2\}$ , $R=\{1,4\}$ ; a function
b) $D=\{3,5,7\}$ , $R=\{1,2,4\}$ ; not a function: $x=3$ gives $y=1$ and $y=4$
c) $D=[-2,\infty)$ , $R=(-\infty,\infty)$ ; not a function: e.g. $x=2$ gives $y=-1$ and $y=3$

a) $5$
b) $x^2-4$
c) $x^2-2x-2$

a) A: $y=|x|$ ; B: $y=\sqrt{x}$ ; C: $y=x^2$ ; D: $y=\frac{1}{x}$
b) $y=x^2$ : $D=(-\infty,\infty)$ , $R=[0,\infty)$ ; $y=\sqrt{x}$ : $D=[0,\infty)$ , $R=[0,\infty)$ ; $y=\frac{1}{x}$ : $D=R=(-\infty,0)\cup(0,\infty)$ ; $y=|x|$ : $D=(-\infty,\infty)$ , $R=[0,\infty)$
c) At $x=3$ : $9$ and $3$ . At $x=\frac{1}{2}$ : $\frac{1}{4}$ and $\frac{1}{2}$ .

a) $D=(-\infty,5]$ , $R=[0,\infty)$
b) $D=(-\infty,-4)\cup(-4,\infty)$ , $R=(-\infty,-1)\cup(-1,\infty)$
c) $D=[-3,3]$ , $R=[-2,1)\cup[3,5]$

a) $f^{-1}(x)=\dfrac{5x-1}{2}$
b) $f^{-1}(3)=7$ ; $\dfrac{1}{f(3)}=\dfrac{5}{7}$
c) $q^{-1}(x)=x^2-3$ ; $D=[0,\infty)$ , $R=[-3,\infty)$

Real World

a) $t=4$ s; $D=[0,4]$ , $R=[0,31.25]$ (metres)
b) $t=0$ : the moment it is tossed. $t=3$ : back at balcony height on the way down.
c) Nothing: $t=-1$ is before the toss, outside the domain $[0,4]$ .

a) $F^{-1}(x)=\dfrac{x-32}{1.8}$ ; input: a temperature in °F; output: the same temperature in °C
b) $30$ °C
c) $D=[-273.15,\infty)$ , $R=[-459.67,\infty)$ ; $F^{-1}(-500)\approx-295.6$ °C, colder than absolute zero, so $-500$ °F is impossible: it is not in the range of $F$ .

a) $C(h)=4h$ for $0\le h\le 5$ ; $C(h)=20$ for $5<h\le 24$ ; $R=[0,20]$
b) $4h=14$ , so $3.5$ hours
c) No: every $h$ in $[5,24]$ costs $20, so $C$ fails the horizontal line test and has no inverse on $[0,24]$ . The inverse $C^{-1}(x)=\frac{x}{4}$ exists only on $[0,5]$ .

Think It Through

a) $f(0)=f(-4)=3$ , so the inverse pairs $3$ with both $0$ and $-4$ : it is not a function.
b) The square root step: $y+2=\pm\sqrt{x+1}$ , so $y=-2\pm\sqrt{x+1}$ .
c) $x\ge-2$ : $f^{-1}(x)=-2+\sqrt{x+1}$ ; $x\le-2$ : $f^{-1}(x)=-2-\sqrt{x+1}$ (both inverses have domain $[-1,\infty)$ )

a) e.g. $y=\dfrac{1}{x-3}-2$
b) e.g. $y=\sqrt{4-x}+1$
c) e.g. $f(x)=\dfrac{1}{x}$ : $f(f(x))=\dfrac{1}{1/x}=x$ ; or $f(x)=5-x$ : $f(f(x))=5-(5-x)=x$

Boss Level

Boss

a) $9$ (use $x=1$ )
b) $f(x)=x^2-x+3$
c) $R=\left[\frac{11}{4},\infty\right)$ ; $k=\frac{11}{4}$

Free cheat sheet

Introduction to Functions Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

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The same page with blanks: fill it in from memory, then check.

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Phone card

The essentials on one screen. Save it to your photos.

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Most testedFind $f^{-1}(x)$ for $f(x)=\dfrac{2x+1}{5}$ , then state the domain and range of $f^{-1}$ . (card 7)

1Function or relation?

UseA function gives each input exactly one output. Graph: no vertical line meets it twice. Mapping: no input has two arrows. Outputs may repeat.

Example $\{(1,4),(2,4),(3,7)\}$ is a function; $\{(4,1),(4,2)\}$ is not.

TrapTest every vertical line. $x=y^2$ passes at $x=-1$ but fails at $x=4$ : $y=\pm2$ .

2Evaluating $f$

UseReplace every $x$ with the input, in brackets. $f(x)$ is the output at $x$ , not $f$ times $x$ .

Example $f(x)=x^2-3x$ : $f(-2)=(-2)^2-3(-2)=10$ ; $f(2a)=4a^2-6a$ .

TrapFor $f(x)=x^2$ , $f(-3)=(-3)^2=9$ , not $-3^2=-9$ : bracket a negative input.

Trap $f(x+1)\ne f(x)+f(1)$ . For $f(x)=x^2$ , $f(x+1)=x^2+2x+1$ .

3Solving $f(x)=k$

Use $f(a)$ : you know the input, find the output. $f(x)=k$ : you know the output, solve for the input(s). $x$ -intercepts: solve $f(x)=0$ .

Example $f(x)=x^2-3x$ , $f(x)=4$ : $x^2-3x-4=0$ , so $x=4$ or $x=-1$ .

Trap $T(e)=0$ asks for the elevation where it is $0^\circ$ C; $T(0)$ is the temperature at sea level.

4Parent functions

$f(x)$	domain	range
$x$	$\mathbb{R}$	$\mathbb{R}$
$x^2$	$\mathbb{R}$	$y\ge0$
$\sqrt{x}$	$x\ge0$	$y\ge0$
$\|x\|$	$\mathbb{R}$	$y\ge0$
$\dfrac{1}{x}$	$x\ne0$	$y\ne0$

Tip $y=x^2$ and $y=|x|$ share a vertex, domain and range, but one is a U and the other a V. $y=1/x$ has asymptotes $x=0$ and $y=0$ .

Trap $\sqrt{(-5)^2}=\sqrt{25}=5$ , not $-5$ : a square root never outputs a negative.

5Finding the domain

UsePolynomial: all real numbers. Square root: radicand $\ge0$ . Rational: denominator $\ne0$ , found before simplifying.

Example $f(x)=\sqrt{10-2x}$ : $-2x\ge-10$ , so $x\le5$ : domain $(-\infty,5]$ .

TrapDividing by a negative flips the sign: $-2x\ge-10$ gives $x\le5$ , not $x\ge5$ .

Trap $(x-2)/(x^2-4)$ excludes $x=2$ and $x=-2$ , even though $x-2$ cancels.

6Range and notation

$\{x\in\mathbb{R}\mid x\ge3\}=[3,\infty)$

UseSquare bracket: included. Round: excluded. $\pm\infty$ always round. Range from a graph: read the $y$ -values, skip gaps: $[-2,1]\cup[3,5]$ . In context, keep only values that make sense.

TrapA thrown ball's domain runs until it lands ( $t=4$ ), not until its peak ( $t=2$ ).

7The inverse $f^{-1}$

$f\left(f^{-1}(x)\right)=x\qquad f^{-1}\left(f(x)\right)=x$

UseSwap $x$ and $y$ , then solve for $y$ . Domain of $f^{-1}$ = range of $f$ , and range of $f^{-1}$ = domain of $f$ .

Example $f(x)=(2x+1)/5$ : $x=(2y+1)/5$ , so $f^{-1}(x)=(5x-1)/2$ .

Trap $f^{-1}(x)$ is the inverse function, not $1/f(x)$ .

8Graph of the inverse

UseReflect over $y=x$ : $(a,b)$ becomes $(b,a)$ . If $f$ fails the horizontal-line test, restrict its domain so $f^{-1}$ is a function.

Example $f(x)=x^2$ , $x\ge0$ : $f^{-1}(x)=\sqrt{x}$ .

TrapReflect over the line $y=x$ , never over an axis: swap the coordinates.

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Core Skills (K)

K1 Function or Not?

K2 Function Notation

K3 Parent Functions

K4 Domain & Range

K5 Inverses

Real World (A)

A1 Balcony Toss

A2 Celsius to Fahrenheit

A3 Parking Cap

Think It Through (T)

T1 Spot the Flaw

T2 Build It Backwards

Boss Level

A Function in Disguise

Answer key

Core Skills

Real World

Think It Through

Boss Level

Introduction to Functions Cheat Sheet

Unit Summary

Build your own

Phone card

1Function or relation?

2Evaluating fff

3Solving f(x)=kf(x)=kf(x)=k

4Parent functions

5Finding the domain

6Range and notation

7The inverse f−1f^{-1}f−1

8Graph of the inverse

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2Evaluating $f$

3Solving $f(x)=k$

7The inverse $f^{-1}$