Transcript L1 and L2 Lesson Powerpoint

Unit 3: An Introduction to Functions
Do Now:
Can you think of real world situations where you would input
something into a machine and get a different output?
For example, a toaster is a machine. When bread is input in the
machine the output is toast.
In the table below list at least two machines along with their inputs
and outputs.
Input
Machine
Output
Bread
toaster
Debit Card
ATM
Cash
Vending
Machine
M&Ms
Money
toast
A function is a rule that assigns each input to exactly one output.
In the do now example, the machine is a function.
You can think of a function as being a box with a special rule,
where the input is the stuff that goes into the box and the
output is the stuff that comes out of the box.
Example 1: Movie Title Box
Put a movie title into the box and the output is the first letter of a movie
title. (Only movie titles can go in.)
Groundhog Day
G
101 Dalmations
???
Domain and Range
The stuff that goes IN the box (the INPUT) is called
the DOMAIN.
The stuff that spits OUT of the box (the OUTPUT) is
called the RANGE.
For the previous example,
the domain is all movie titles that start with a letter
(101 Dalmations is not in the domain.)
And the range is all letters of the alphabet.
(there is a movie title for each letter of the alphabet!)
Example 2: Now let’s try a box with a rule of adding 3. Draw a diagram and
choose several values to put into the box to complete the table. State the
domain and range.
Domain: all real numbers
Range: all real numbers
Example 3: Create a table of input and output values for the
function rule 3x – 5. Then state the domain and range of the
function.
Input
0
Output
-5
1
-2
-4
.5
-17
-3.5
Domain = All real numbers
Range =
All real numbers
1
Example 3: Create a table of input and output values for the function rule .
𝑥
Then state the domain and range of the function.
Input
1
Output
1
0
UNDEFINED
Domain =
All real numbers
except 0
Range =
All real numbers
Example 4: Create a table of input and output values for the function rule
Then state the domain and range of the function.
Input
1
Output
-1
2
UNDEFINED
1
.
𝑥−2
Domain =
All real numbers
except 2
Range =
All real numbers
Different Representations of Relations and Functions
A. Mapping Diagrams:
2.)
1.)
Function
Domain
Range
Yes
No
{-2,2,4,5,6}
{4,16,25,36}
Function
Yes
Domain
{-2,0,1,3}
Range
{-3,1,3,4}
No
B. Ordered pairs:
1.) {(-3, 2) (-1, 5) (2, 9) (-3, 0)
Function
2.) {(1, 3) (2, 7) (6, 11) (21, 53)}
Function
Domain
Yes No
{-3,-1, 2}
Yes
No
Domain
{1,2,6,21)
Range
{0, 2, 5, 9}
Range
{3,7,11,53}
Lesson 1: Day 2
Using Function Notation and Evaluating Functions
The function f acts upon the
input, x, which we put within
parentheses. The output is
shown on the other side of the
= symbol.
For example, f(bread)= toast (where the function is the toaster)
f(Frozen) = F
f(Bourne Identity) = B
To describe a function that can have an unlimited number of inputs,
we must describe f using an equation.
We use the letter x to represent the inputs (also the domain)
This means the letter x is said to be the independent variable.
The output (also the range) is typically represented by the letter y and is
said to be the dependent variable (simply because its value depends on the
value of the independent variable).
Using the rule “add 5”, let x be the input.
Then:
f(x) = x + 5
x goes in, x + 5 comes out.
f(x) is the official output name
Now let’s do some examples using function notation.
a.) If the input is 4, what is the output?
f(4) = 4 + 5
f(4) = 9
b.) Find f(-2).
f(-2) = -2 + 5
f(-2) = 3
Guided Practice
Exercises:
For each of the problems below,
a) Write a function using function notation to model the description.
b) Evaluate the following for each: f(0), f(-3) and f(7).
c) State the domain and range of the function.
1.) Three times a number plus 8.
a.) f(x) = 3x + 8
b.) f(0) = 8
f(-3) = -1
f(7) = 29
c.) domain = all real numbers
range = all real numbers
2.) The square of a number minus 4.
a.) f(x) = x2 - 4
b.) f(0) = -4
f(-3) = 5
f(7) = 45
c.) domain = all real numbers
range = all real numbers
3.) Twice a number subtracted from 36.
a.) f(x) = 36 - 2x
b.) f(0) = 36
f(-3) = 42
f(7) = 22
c.) domain = all real numbers
range = all real numbers
4.) Six more than four times a number.
a.) f(x) = 4x + 6
b.) f(0) = 6
f(-3) = -6
f(7) = 34
c.) domain = all real numbers
range = all real numbers
5.) Thirteen less than a number.
a.) f(x) = x - 13
b.) f(0) = -13
f(-3) = -16
f(7) = -6
c.) domain = all real numbers
range = all real numbers
Real World applications of Domain and Range
6.) Does the relationship represent a function? Explain.
A. {students in a school} → {locker}
Yes
B. {telephone number } → {students in a school}
C. {cars in a parking lot } → {license plate number}
D. {U.S citizen who works} → {social security #}
No
Yes
Yes
E. {Year} → {Total number of North Atlantic hurricanes}
Complete Function Tables Worksheet
Yes
Sometimes functions have some meaning and a name for them
is chosen to reflect that meaning.
For example, the temperature of a heated object varies depending
on how long it has been removed from the heat source.
Suppose T is the name of the function that describes the
Temperature of a hamburger that is cooling.
We use the lowercase t, standing for time, as the independent
variable since the Temperature, T, of the hamburger depends
on the time, t.
Suppose the Temperature function is shown below.
Time (t)
T
Temperature (T)
0 minutes
170° F
1 minute
169°F
2 minutes
167°F
3 minutes
165°F
The Temperature
function tells us that
the moment the
hamburger leaves the
grill, it is 170° F.
After 1 minute, the
hamburger is 169° F.
After 2 minutes, the
hamburger is 167° F
and so forth.
a.) What is the value of T(3 minutes)?
165˚F
b.) Suppose the Temperature function is given by the equation
T(t) = 170 – 3t
where t varies from 0 to 25 minutes. What is the value of T(0), T(1),
and T(10)?
T(0) = 170˚F
T(1) = 167˚F
T(10) = 140˚F
c.) What is the value of T at t = 0, 1, and 2 if the equation of T is given by
T(t) = 3t + 20?
T(0) = 20˚F
T(1) = 23˚F
T(2) = 26˚F
Homework: Complete problem set in packet