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1. 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.

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. Linear or exponential: equal differences or equal factors

Warm-up

Three quick questions from earlier lessons.

Discover

Offer A: 1,000 dollars a day for 30 days. Offer B: 1 cent the first day, then double every day. Which pays more on day 30?

Offer B
Offer A

Key idea

Linear adds, exponential multiplies.

Over equal steps in

x

, a linear function changes by equal differences (add

m

); an exponential function changes by equal factors (multiply by

b

). A growing exponential eventually overtakes any linear function.

Worked example

$y$ : 3, 6, 12, 24.

Not linear

Differences 3, 6, 12: not equal.

Exponential, factor 2

Ratios 2, 2, 2.

24 \cdot 2 = 48

Next value.

Your turn, guided

Your turn: an exponential function has $g(0) = 5$ , $g(1) = 15$ . Find $g(2)$ .

45

Factor 3.

Practice ★

A table gives $x = 0, 1, 2, 3$ and $y = 4, 12, 36, 108$ . What kind of relationship is it?
1. Exponential: $y$ is multiplied by 3 at each step
2. Linear: $y$ goes up by 8 at each step
3. Neither
A table gives $x = 0, 1, 2, 3$ and $y = 8, 24, 72, 216$ . What kind of relationship is it?
1. Exponential: $y$ is multiplied by 3 at each step
2. Linear: $y$ goes up by 16 at each step
3. Neither

An exponential function has $g(0) = 9$ and $g(1) = 18$ . Find $g(3)$ .

Watch out!

Practice ★★

$f(x) = 57x + 380$ and $g(x) = 1.04^x$ . For very large $x$ , which is greater?
1. $g$ : exponential growth eventually overtakes linear growth
2. $f$ : it is greater now and grows faster
3. They stay about equal
$f(x) = 73x + 397$ and $g(x) = 1.04^x$ . For very large $x$ , which is greater?
1. $g$ : exponential growth eventually overtakes linear growth
2. $f$ : it is greater now and grows faster
3. They stay about equal

Exit ticket

Three questions to check you have got it.

An exponential function has $g(0) = 9$ and $g(1) = 18$ . Find $g(3)$ .
An exponential function has $g(0) = 4$ and $g(1) = 20$ . Find $g(3)$ .
An exponential function has $g(0) = 5$ and $g(1) = 15$ . Find $g(3)$ .

Summary card

Linear or exponential?

Equal differences: linear. Equal factors: exponential.

Watch out: exponential growth wins in the end.

4. Write an exponential function from a description or two points

Warm-up

Three quick questions from earlier lessons.

Discover

A table shows a function: $f(0) = 4$ , $f(1) = 12$ , $f(2) = 36$ , $f(3) = 108$ .

What do you multiply by to go from one value to the next? Can you write one formula that gives all four values?

Key idea

An exponential function can be written

f(x) = a \cdot b^x

, where

a = f(0)

and

b

is the factor for each increase of 1 in

x

If the quantity is multiplied by $b$ every $n$ units instead, divide the input by $n$ : $f(x) = a \cdot b^{\frac{x}{n}}$ .

Worked example

Write $f$ for the table above.

a = 4

The starting value is f(0).

b = 3

Divide consecutive values: 12 ÷ 4 = 3.

f(x) = 4 \cdot 3^x

Put them in the form a · b^x.

Your turn, guided

Your turn: $f(0) = 5$ and $f(1) = 10$ . Fill in the missing line.

a = 5

b = 2

10 ÷ 5 = 2.

f(x) = 5 \cdot 2^x

Use the form a · b^x.

Practice ★

Plot the points of the graph of $f(x) = 4 \cdot 2^x$ at $x = 0$ and $x = 1$ .
Plot the points of the graph of $f(x) = 3 \cdot 4^x$ at $x = 0$ and $x = 1$ .

An exponential function $f$ has $f(0) = 9$ , and each time $x$ increases by 1, the value of $f$ is multiplied by 5. Write $f(x)$ .
An exponential function $f$ has $f(0) = 9$ , and each time $x$ increases by 1, the value of $f$ is multiplied by 4. Write $f(x)$ .
An exponential function $f$ has $f(0) = 6$ , and each time $x$ increases by 1, the value of $f$ is multiplied by 2. Write $f(x)$ .
An exponential function $f$ has $f(0) = 6$ , and each time $x$ increases by 1, the value of $f$ is multiplied by 4. Write $f(x)$ .
An exponential function $f$ has $f(0) = 4$ , and each time $x$ increases by 1, the value of $f$ is multiplied by 2. Write $f(x)$ .

Watch out!

Practice ★★

A population is 7 thousand at $t = 0$ and is multiplied by 3 every 4 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 8 thousand at $t = 0$ and is multiplied by 3 every 3 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 4 thousand at $t = 0$ and is multiplied by 2 every 3 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 8 thousand at $t = 0$ and is multiplied by 2 every 2 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 9 thousand at $t = 0$ and is multiplied by 3 every 5 years. Write the population $P(t)$ , in thousands, after $t$ years.

Exit ticket

Three questions to check you have got it.

A population is 7 thousand at $t = 0$ and is multiplied by 3 every 4 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 8 thousand at $t = 0$ and is multiplied by 3 every 3 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 4 thousand at $t = 0$ and is multiplied by 2 every 3 years. Write the population $P(t)$ , in thousands, after $t$ years.

Summary card

Writing an exponential function

$f(x) = a \cdot b^x$ : $a = f(0)$ , $b = \dfrac{f(1)}{f(0)}$ .
Multiplied by $b$ every $n$ units: $f(x) = a \cdot b^{\frac{x}{n}}$ .

Watch out: the starting value is $f(0)$ , and it goes in front, not in the base.

Answers: Comparing functions

1. Domain and range in interval notation

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

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

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

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

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. Linear or exponential: equal differences or equal factors

Exponential: $y$ is multiplied by 3 at each step
Exponential: $y$ is multiplied by 3 at each step

$72$

$g$ : exponential growth eventually overtakes linear growth
$g$ : exponential growth eventually overtakes linear growth

$72$
$500$
$135$

4. Write an exponential function from a description or two points

$(0, 4),\ (1, 8)$
$(0, 3),\ (1, 12)$

$9 \cdot 5^{x}$
$9 \cdot 4^{x}$
$6 \cdot 2^{x}$
$6 \cdot 4^{x}$
$4 \cdot 2^{x}$

$7 \cdot 3^{\frac{t}{4}}$
$8 \cdot 3^{\frac{t}{3}}$
$4 \cdot 2^{\frac{t}{3}}$
$8 \cdot 2^{\frac{t}{2}}$
$9 \cdot 3^{\frac{t}{5}}$

$7 \cdot 3^{\frac{t}{4}}$
$8 \cdot 3^{\frac{t}{3}}$
$4 \cdot 2^{\frac{t}{3}}$