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1. Function notation: evaluate and read f(x)

Discover

$f(x) = x + 5$ is a rule: it takes a number and adds 5.

What does $f(2)$ mean?

The output of $f$ when the input is 2
$f$ times 2
The input that gives the output 2

Key idea

A function gives exactly one output for each input. We write the output as $f(x)$ .

f(a)

is the output of

f

for the input

a

. To find it, replace every

x

in the rule by

(a)

, with parentheses.

On a graph, $f(a)$ is the height of the graph above $x = a$ .

Worked example

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

f(-2) = 3(-2)^2 - (-2)

Replace each x by (−2), with the parentheses.

= 3 \cdot 4 + 2

(−2)² = 4, and subtracting −2 adds 2.

= 14

Multiply, then add.

Your turn, guided

Your turn: $f(x) = 2x^2 + 1$ . Fill in the missing line.

f(-3) = 2(-3)^2 + 1

Replace x by (−3).

f(-3) = 19

(−3)² = 9, then 2 · 9 + 1.

Practice ★

$f(x) = 2x^2 + 3$ . Find $f(-3)$ .
$f(x) = 4x^2 + 8$ . Find $f(-2)$ .
$f(x) = 4x^2 + 2$ . Find $f(-5)$ .
$f(x) = 5x^2 + 5$ . Find $f(-4)$ .
$f(x) = 3x^2 + 3$ . Find $f(-4)$ .

Watch out!

Practice ★★

$f(x) = x^2 + 3$ . Find $f(2 + 3)$ .
$f(x) = x^2 + 2$ . Find $f(4 + 4)$ .
$f(x) = x^2 + 7$ . Find $f(4 + 1)$ .
$f(x) = x^2 + 3$ . Find $f(3 + 3)$ .

Here is the graph of $y = f(x)$ . Tap the point of the graph that shows $f(2)$ .
Here is the graph of $y = f(x)$ . Tap the point of the graph that shows $f(2)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = x^2 + 3$ . Find $f(2 + 3)$ .
$f(x) = x^2 + 2$ . Find $f(4 + 4)$ .
$f(x) = x^2 + 7$ . Find $f(4 + 1)$ .

Summary card

Function notation

$f(a)$ is the output for the input $a$ .
Replace every $x$ by $(a)$ , with parentheses: $f(-2)$ for $x^2$ is $(-2)^2 = 4$ .
On a graph: the height above $x = a$ .

Watch out: $f(a + b)$ is not $f(a) + f(b)$ .

2. Average rate of change; where a function increases or decreases

Warm-up

Three quick questions from earlier lessons.

Discover

A car drives 120 miles in 2 hours. What is its average speed?

60 miles per hour
$\tfrac{1}{60}$ hour per mile
120 miles per hour

Key idea

The average rate of change tells how fast the output changes, per unit of input.

The average rate of change of

f

from

x = a

x = b

\dfrac{f(b) - f(a)}{b - a}

, the slope of the line through the two points of the graph.

$f$ is increasing on an interval if its graph goes up from left to right there, and decreasing if it goes down.

Worked example

Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 4$ .

f(4) - f(1) = 16 - 1 = 15

Change in the output.

4 - 1 = 3

Change in the input.

Rate

= \dfrac{15}{3} = 5

Divide: output change per unit of input.

Your turn, guided

Your turn: $f(x) = 2x^2$ , from $x = 0$ to $x = 3$ .

f(3) - f(0) = 18

f(3) = 18, f(0) = 0.

Change in

x

3

3 − 0.

Rate

= 6

18 ÷ 3.

Practice ★

Find the average rate of change of $f(x) = 3x^2$ from $x = 0$ to $x = 6$ .
Find the average rate of change of $f(x) = 4x^2$ from $x = 1$ to $x = 5$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 6$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 4$ .

Watch out!

Practice ★★

A tank holds 16 liters at time 0 and $61$ liters after 5 minutes. What is the average rate of change of the volume, in liters per minute?
A tank holds 29 liters at time 0 and $49$ liters after 4 minutes. What is the average rate of change of the volume, in liters per minute?
A tank holds 12 liters at time 0 and $36$ liters after 6 minutes. What is the average rate of change of the volume, in liters per minute?

$f(x) = -(x - 5)^2 + 8$ . On which interval is $f$ increasing?
1. $(-\infty, 5)$
2. $(5, \infty)$
3. $(-\infty, 8)$
$f(x) = -(x - 1)^2 + 7$ . On which interval is $f$ increasing?
1. $(-\infty, 1)$
2. $(1, \infty)$
3. $(-\infty, 7)$
$f(x) = -(x - 3)^2 + 2$ . On which interval is $f$ increasing?
1. $(-\infty, 3)$
2. $(3, \infty)$
3. $(-\infty, 2)$

Exit ticket

Three questions to check you have got it.

Find the average rate of change of $f(x) = 3x^2$ from $x = 0$ to $x = 6$ .
Find the average rate of change of $f(x) = 4x^2$ from $x = 1$ to $x = 5$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 6$ .

Summary card

Average rate of change

From $a$ to $b$ : $\dfrac{f(b) - f(a)}{b - a}$ , output change over input change.
Increasing: the graph rises from left to right.
Give intervals of increase with $x$ -values.

Watch out: change in $f(x)$ on top, change in $x$ below.

3. Domain and range in interval notation

Warm-up

Three quick questions from earlier lessons.

Discover

Which numbers can you put into $f(x) = \sqrt{x}$ ?

0 and the positive numbers
Every number
Only the positive numbers, not 0

Key idea

The domain is the set of inputs a function accepts. The range is the set of outputs it gives.

Leave out of the domain every

x

that makes a denominator

0

or puts a negative number under a square root. A square bracket includes the end:

[2, \infty)

contains 2; a parenthesis leaves it out:

(2, \infty)

does not.

Worked example

Find the domain of $f(x) = \sqrt{x + 2}$ .

x + 2 \ge 0

The inside of a square root must not be negative.

x \ge -2

Subtract 2 from both sides.

Domain:

[-2, \infty)

−2 is allowed (√0 = 0), so use a square bracket.

Your turn, guided

Your turn: $f(x) = \sqrt{x - 5}$ . Fill in the missing line.

x - 5 \ge 0

The inside must not be negative.

The domain starts at

x = 5

[5, \infty)

Add 5 to both sides: the domain starts at this number.

Practice ★

What is the domain of $f(x) = \sqrt{x - 9}$ ? Give it as an interval.
What is the domain of $f(x) = \sqrt{x - 5}$ ? Give it as an interval.
What is the domain of $f(x) = \sqrt{x - 6}$ ? Give it as an interval.
What is the domain of $f(x) = \sqrt{x - 4}$ ? Give it as an interval.

Watch out!

Practice ★★

What is the domain of $f(x) = \dfrac{ 9 }{x - 6}$ ? Give it as an interval or a union of intervals.
What is the domain of $f(x) = \dfrac{ 6 }{x - 5}$ ? Give it as an interval or a union of intervals.
What is the domain of $f(x) = \dfrac{ 4 }{x - 3}$ ? Give it as an interval or a union of intervals.

What is the range of $f(x) = (x - 5)^2 + 9$ ? Give it as an interval.
What is the range of $f(x) = (x - 3)^2 + 6$ ? Give it as an interval.
What is the range of $f(x) = (x - 5)^2 + 4$ ? Give it as an interval.

Exit ticket

Three questions to check you have got it.

What is the range of $f(x) = (x - 5)^2 + 9$ ? Give it as an interval.
What is the range of $f(x) = (x - 3)^2 + 6$ ? Give it as an interval.
What is the range of $f(x) = (x - 5)^2 + 4$ ? Give it as an interval.

Summary card

Domain and range

Domain: the inputs. Leave out zero denominators and negatives under a square root.
Range: the outputs.
$[a, \infty)$ includes $a$ ; $(a, \infty)$ does not.

Watch out: the range uses $y$ -values, not the $x$ of a vertex.

4. Piecewise-defined functions and absolute value

Warm-up

Three quick questions from earlier lessons.

Discover

A parking lot charges 5 dollars for up to 2 hours, and 9 dollars for a longer stay. What does a stay of exactly 2 hours cost?

5 dollars
9 dollars

Key idea

A piecewise function uses a different formula on different parts of its domain.

To find

f(a)

: first find which condition

a

satisfies, then use that piece's formula. Check the boundary carefully (

<

\le

The absolute value is piecewise: $|x| = x$ if $x \ge 0$ , and $|x| = -x$ if $x < 0$ .

Worked example

$f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x + 3 & \text{if } x \ge 1 \end{cases}$ . Find $f(1)$ .

Use the second piece,

2x + 3

Check the conditions first: 1 is not less than 1, but 1 ≥ 1.

f(1) = 2 \cdot 1 + 3 = 5

Replace x by 1.

Your turn, guided

Your turn, with the same $f$ : find $f(-2)$ .

Use the first piece,

x^2

−2 < 1.

f(-2) = 4

(−2)² = 4.

Practice ★

$f(x) = \begin{cases} x + 4 & \text{if } x < 8 \\ 4x & \text{if } x \ge 8 \end{cases}$ . Find $f(0)$ .
$f(x) = \begin{cases} x + 8 & \text{if } x < 7 \\ 4x & \text{if } x \ge 7 \end{cases}$ . Find $f(2)$ .
$f(x) = \begin{cases} x + 7 & \text{if } x < 3 \\ 3x & \text{if } x \ge 3 \end{cases}$ . Find $f(1)$ .
$f(x) = \begin{cases} x + 6 & \text{if } x < 6 \\ 4x & \text{if } x \ge 6 \end{cases}$ . Find $f(1)$ .

Watch out!

Practice ★★

$f(x) = \begin{cases} x + 3 & \text{if } x < 5 \\ 3x & \text{if } x \ge 5 \end{cases}$ . Find $f(5)$ .
$f(x) = \begin{cases} x + 4 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . Find $f(3)$ .
$f(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 5x & \text{if } x \ge 1 \end{cases}$ . Find $f(1)$ .
$f(x) = \begin{cases} x + 2 & \text{if } x < 5 \\ 2x & \text{if } x \ge 5 \end{cases}$ . Find $f(5)$ .

For $x < 7$ , which expression is equal to $|x - 7|$ ?
1. $7 - x$
2. $x - 7$
3. $-x - 7$
For $x < 5$ , which expression is equal to $|x - 5|$ ?
1. $5 - x$
2. $x - 5$
3. $-x - 5$

Exit ticket

Three questions to check you have got it.

$f(x) = \begin{cases} x + 3 & \text{if } x < 5 \\ 3x & \text{if } x \ge 5 \end{cases}$ . Find $f(5)$ .
$f(x) = \begin{cases} x + 4 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . Find $f(3)$ .
$f(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 5x & \text{if } x \ge 1 \end{cases}$ . Find $f(1)$ .

Summary card

Piecewise functions

Find the piece whose condition the input satisfies.
Use that piece's formula.
$|x| = x$ for $x \ge 0$ , and $-x$ for $x < 0$ .

Watch out: at a boundary, read $<$ and $\ge$ carefully.

5. Inverse functions

Warm-up

Three quick questions from earlier lessons.

Discover

$f(x) = x + 7$ adds 7. Which function undoes it, and gives back the number you started with?

$x - 7$
$\dfrac{1}{x + 7}$
$x + 7$

Key idea

The inverse function $f^{-1}$ undoes $f$ : it swaps inputs and outputs.

f(a) = b

, then

f^{-1}(b) = a

. To find

f^{-1}(x)

: write

y = f(x)

, solve for

x

, then rename. The graph of

f^{-1}

is the reflection of the graph of

f

across the line

y = x

$f^{-1}(x)$ is not $\dfrac{1}{f(x)}$ : the $-1$ is not an exponent here.

Worked example

$f(x) = 3x - 5$ . Find $f^{-1}(x)$ .

y = 3x - 5

Name the output y.

y + 5 = 3x

f subtracts 5 last, so undo that first: add 5.

x = \dfrac{y + 5}{3}

Then undo × 3: divide by 3.

f^{-1}(x) = \dfrac{x + 5}{3}

Rename the input x.

Your turn, guided

Your turn: $f(x) = 2x + 6$ .

y = 2x + 6

Name the output y.

y - 6 = 2x

Undo + 6 first.

f^{-1}(x) = \dfrac{x - 6}{2}

Divide by 2, then rename.

Practice ★

$f(x) = 7x + 9$ . Find $f^{-1}(x)$ .
$f(x) = 2x + 2$ . Find $f^{-1}(x)$ .
$f(x) = 7x + 5$ . Find $f^{-1}(x)$ .
$f(x) = 2x + 9$ . Find $f^{-1}(x)$ .

Watch out!

Practice ★★

$f(x) = 2x + 6$ . Find $f^{-1}(10)$ .
$f(x) = 2x + 9$ . Find $f^{-1}(11)$ .
$f(x) = 3x + 1$ . Find $f^{-1}(10)$ .

The point $(2, 7)$ is on the graph of $f$ . Tap the point that must be on the graph of $f^{-1}$ .
The point $(5, 7)$ is on the graph of $f$ . Tap the point that must be on the graph of $f^{-1}$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 7x + 9$ . Find $f^{-1}(x)$ .
$f(x) = 2x + 2$ . Find $f^{-1}(x)$ .
$f(x) = 7x + 5$ . Find $f^{-1}(x)$ .

Summary card

Inverse functions

Write $y = f(x)$ .
Solve for $x$ , undoing the last operation first.
Rename: $f^{-1}(x)$ . Check that $f(f^{-1}(x)) = x$ .

Watch out: $f^{-1}(x)$ is not $\dfrac{1}{f(x)}$ .

6. Transformations: shifts and reflections

Warm-up

Three quick questions from earlier lessons.

Discover

Move the sliders and watch the graph of $y = (x - h)^2 + k$ .

The graph of y = (x − h)² + k, a parabola whose vertex is at (h, k) and moves with the sliders.

When $h$ goes from 0 to 2, which way does the graph move?

To the right
To the left

Key idea

Changing the rule moves or flips the graph without changing its shape.

y = f(x - h) + k

moves the graph of

f

h

units to the right and

k

units up.

y = -f(x)

reflects it across the

x

-axis;

y = f(-x)

reflects it across the

y

-axis.

A change inside $f(\dots)$ acts on $x$ (horizontally); a change outside acts on $y$ (vertically).

Worked example

Describe $y = (x + 1)^2 - 3$ compared with $y = x^2$ .

1 unit to the left

x + 1 = x − (−1): h = −1.

3 units down

− 3 outside: k = −3.

Vertex:

(-1, -3)

The vertex (0, 0) moves with the graph.

Your turn, guided

Your turn: $y = (x - 4)^2 + 2$ .

4 units to the right

x − 4: h = 4.

2 units up

+ 2 outside.

The vertex is at

x = 4

The vertex moves from x = 0 to x = …

Practice ★

The graph of $y = x^2$ is moved 1 units to the right and 6 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 3 units to the right and 1 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 4 units to the right and 6 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 7 units to the right and 1 units down. Write the equation of the new graph.

Watch out!

Practice ★★

Draw the graph of $y = (x - 1)^2 - 3$ : place the vertex and one more point.
Draw the graph of $y = (x - 1)^2 - 1$ : place the vertex and one more point.

Match each equation with what it does to the graph of $y = f(x)$ .
1. $y = f(x) + 3$
2. $y = f(x - 3)$
3. $y = -f(x)$
4. $y = f(-x)$
1. moves it up 3 units
2. moves it 3 units to the right
3. reflects it across the x-axis
4. reflects it across the y-axis

Exit ticket

Three questions to check you have got it.

The graph of $y = x^2$ is moved 1 units to the right and 6 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 3 units to the right and 1 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 4 units to the right and 6 units down. Write the equation of the new graph.

Summary card

Shifts and reflections

$f(x - h)$ : $h$ units right. $f(x) + k$ : $k$ units up.
$-f(x)$ : flip across the $x$ -axis. $f(-x)$ : flip across the $y$ -axis.
Inside $f(\dots)$ acts on $x$ ; outside acts on $y$ .

Watch out: $f(x + 3)$ moves the graph left.

7. Sequences: explicit and recursive formulas

Warm-up

Three quick questions from earlier lessons.

Discover

A sequence starts $5, 8, 11, 14, \dots$ Which rule gives the next term from the one before?

Add 3
Multiply by 3

Key idea

A sequence is a function whose inputs are the whole numbers $1, 2, 3, \dots$ : $a_n$ is its $n$ th term.

An explicit formula gives

a_n

from

n

directly (for example

a_n = 3n + 2

). A recursive formula gives a first term and a rule from one term to the next (for example

a_1 = 5

a_{n+1} = a_n + 3

Worked example

$a_1 = 2$ and $a_{n+1} = 3a_n - 1$ . Find $a_4$ .

a_2 = 3 \cdot 2 - 1 = 5

Apply the rule to a₁.

a_3 = 3 \cdot 5 - 1 = 14

Then to a₂.

a_4 = 3 \cdot 14 - 1 = 41

Then to a₃.

Your turn, guided

Your turn: $a_n = n^2 - 1$ . Find $a_5$ .

a_5 = 5^2 - 1

Replace n by 5.

a_5 = 24

25 − 1.

Practice ★

$a_n = n^2 + 5$ . Complete the table.
$a_n = n^2 + 9$ . Complete the table.

Watch out!

Practice ★★

$a_1 = 3$ and $a_{n+1} = 2a_n - 7$ . Find $a_3$ .
$a_1 = 6$ and $a_{n+1} = 2a_n - 3$ . Find $a_3$ .
$a_1 = 4$ and $a_{n+1} = 2a_n - 6$ . Find $a_3$ .

Write a formula for $a_n$ , starting at $n = 1$ : $1,6, 11, 16, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $3,10, 17, 24, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8,14, 20, 26, \dots$

Exit ticket

Three questions to check you have got it.

Write a formula for $a_n$ , starting at $n = 1$ : $1,6, 11, 16, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $3,10, 17, 24, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8,14, 20, 26, \dots$

Summary card

Sequences

Explicit: $a_n$ straight from $n$ .
Recursive: a first term and a rule from $a_n$ to $a_{n+1}$ .

Watch out: check your formula with $n = 1$ .

Answers: Functions

1. Function notation: evaluate and read f(x)

$21$
$24$
$102$
$85$
$51$

$28$
$66$
$32$
$39$

$(2, -1)$
$(2, 3)$

$28$
$66$
$32$

2. Average rate of change; where a function increases or decreases

$18$
$24$
$7$
$5$

$(-\infty, 5)$
$(-\infty, 1)$
$(-\infty, 3)$

$18$
$24$
$7$

3. Domain and range in interval notation

$[9, \infty)$
$[5, \infty)$
$[6, \infty)$
$[4, \infty)$

$(-\infty, 6) \cup (6, \infty)$
$(-\infty, 5) \cup (5, \infty)$
$(-\infty, 3) \cup (3, \infty)$

$[9, \infty)$
$[6, \infty)$
$[4, \infty)$

$[9, \infty)$
$[6, \infty)$
$[4, \infty)$

4. Piecewise-defined functions and absolute value

$4$
$10$
$8$
$7$

$15$
$6$
$5$
$10$

$7 - x$
$5 - x$

$15$
$6$
$5$

5. Inverse functions

$\frac{x}{7} - \frac{9}{7}$
$\frac{x}{2} - 1$
$\frac{x}{7} - \frac{5}{7}$
$\frac{x}{2} - \frac{9}{2}$

$(7, 2)$
$(7, 5)$

$\frac{x}{7} - \frac{9}{7}$
$\frac{x}{2} - 1$
$\frac{x}{7} - \frac{5}{7}$

6. Transformations: shifts and reflections

$\left(x - 1\right)^{2} - 6$
$\left(x - 3\right)^{2} - 1$
$\left(x - 4\right)^{2} - 6$
$\left(x - 7\right)^{2} - 1$

$y = \left(x - 1\right)^{2} - 3$
$y = \left(x - 1\right)^{2} - 1$

$y = f(x) + 3$ → moves it up 3 units · $y = f(x - 3)$ → moves it 3 units to the right · $y = -f(x)$ → reflects it across the x-axis · $y = f(-x)$ → reflects it across the y-axis

$\left(x - 1\right)^{2} - 6$
$\left(x - 3\right)^{2} - 1$
$\left(x - 4\right)^{2} - 6$

7. Sequences: explicit and recursive formulas

$6, 9, 14, 21$
$10, 13, 18, 25$

$-9$
$15$
$-2$

$5 n - 4$
$7 n - 4$
$6 n + 2$

$5 n - 4$
$7 n - 4$
$6 n + 2$