Answer: -3 if x ≥ -4 f(x) = -x if x ≤ -1. Answer: -1 if x ≥ 2 f(x) = x 2 if x < 2

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

Answer: -3 if x ≥ -4f(x) =

-x if x ≤ -1

Answer: -1 if x ≥ 2f(x) =

x2 if x < 2

SWBAT… define a function, learn function notation, and a evaluate function

Agenda 1. Lots of notes on functions with many practice problems (40 min)2. OYO Problems (10 min)

Warm-Up:

1. Set up your notes – Topic is “Functions”

HW#6: Page 1

Wed, 10/31

Happy Halloween!

Ms. Sophia Papaefthimiou

Infinity HS

ObjectivesToday:

1. To define a function

2. To learn function notation

3. To evaluate functions

Tomorrow:

1. To learn function mapping

2. To conduct the vertical line test

3. To find the domain and range of a function

4. To write a function as an ordered pair

What is a function?

A function is like a machine: it has an input and an output. And the output is related somehow to the input.

Function Notation The most common name is "f", but you can have

other names like "g" What goes into the function (the input) is put inside

parentheses after the name of the function Example: f(x)

(pronounced “f of x”) shows you the function is called "f", and "x" goes in.

Question: What if a function was called “g” and “a” went into it? How would you write the function?

y f x • Output Value• Range• Dependent Variable

• Input Value• Domain• Independent Variable

Name of the function

Function Notation: The Symbolic Form

The output is y = f(x)

Function Notation

Function notation replaces the ___ in an equation with ___ Example: Given y = 3x + 2, write the equation in function

notation

f(x) = 3x + 2

Question: Write y = x2 in function notation.

f(x) = x2

y f(x)

Function Notation You used to say “y = 2x + 3; find the value of y when x = -1”

y = 2x + 3

y = 2(-1) + 3

y = -2 + 3

y = 1

Now you say “f(x) = 2x + 3; find f(-1)”

1

-1

f(-1) = 2x + 3

A function P is defined as follows:For x > 0, P(x) = x5 + x4 – 36x – 36For x < 0, P(x) = -x5 + x4 + 36x – 36

What is the value of P(-1)?A. -70B. -36C. 0D. 36

Evaluating functions: Directions:

If f(x) = 2x – 4 and g(x) = x² – 4x, find each value:

1. f(-3)

2. f(3x)

3. g(t)

4. f(q + 1)

5. f(2) + g(-2) 6. f(g(-2)) (Hint: Start from the inside out. Find g(-2) first)

Revisit our objectives

Today:

1. To define a function

2. To learn function notation

3. To evaluate functions

SWBAT… use the vertical line test

Agenda 1. WU (20 min)2. Review HW#6 – page 1 (25 min)

Warm-Up:

Let f(t) be the number of people, in millions, who own cell phones t years after 1990. Explain the meaning of the following statements.

1. f(10) = 100.3

2. f(a) = 20

3. f(20) = b

4. n = f(t)

Review your notes/practice problems

Wed, 10/30

Solution: Cell phones1. f(10) = 100.3: The number of people who own cell phones in the year 2000 is 100,300,000 .

2. f(a) = 20: There are 20,000,000 people who own cell phones a years after 1990.

3. f(20) = b: There will be b million people who own cell phones in the year 2010.

4. n = f(t): The number n is the number of people (in millions) who own cell phones t years after 1990.

Review HW#6 – Page 1

ObjectivesToday

1. To learn function mapping

2. To conduct the vertical line test

3. To find the domain and range of a function

4. To write a function as an ordered pair

Function Mapping

A set of points or equation where every input has exactly one output.

In other words, the domain or x value can not be repeated

This is a function! There is only one arrow coming from each x.In other words, x can not be repeated

This is a function! There is only one arrow coming from each xThere is only one y for each x. It just so happens that it's always the same y for each x.

Function Mapping (cont’d)

This one is not a function.There are two arrows coming from the number 1.The number 1 is associated with two different range elements. In order words, x is repeated.

Function Mapping (cont’d)

Vertical Line Test

No mater where we drop a vertical line, if the vertical line only hits the graph once, it is a function.

So, this graph is a function!

Draw a graph, that would NOT pass the vertical line test.

Vertical Line Test (cont’d)

Intersect at two points These graphs are not functions

Domain and Range

• Domain: What can go into a function. The set of all x values in a function. How “wide” the function is.

• Range: What comes out of a function. The set of all y values in a function. How “tall” the function is.

The domain is the set of all real numbers.The range is y ≥ 0.

SWBAT… list the domain and range of functions

Agenda 1. WU (10 min)2. 10 practice problems (30 min)3. Review HW#6 – Page 1 – 3 (10 min)

Warm-Up:

1. Write your HW in your planner.2. What is the domain and range of f(x) = x + 1?

HW#6: Functions (page 4 – quiz grade)

Mon, 11/5

What is the domain and range of y = x + 1?

Domain: All real numbers

Range: All real numbers2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

Domain and Range (from Friday)

f(x) = x2 – 2 The domain is the set of all real numbers.The range is y ≥ -2.

f(x) = -3x + 10 and g(x) = 2x

1.Solve f(x) = 0

2.Solve f(x) > 0

3.Solve f(x) = g(x)

4.Solve f(x) < g(x)

Question 1/10

Question: For the function f(x) = x2, if the domain is {1, 2, 3}, what is the range?

Question 2/10

Given f(x) = 3x – 5 and the domain is {0, 2, -1}

find the range

Question 3/10

If f(x) = -x2 find a. f(3) b. f(-3)

Question 4/10

Function f is defined by f(x) = -2x2 + 6x – 3

1. Find f(-2)

2. Write as an ordered pair

Question 5/10

Does the diagram represent a function? Why?

24

3

15

0

1

7

Question 6/10

Suppose g(x) = 2x and f(x) = 4x.

What is g(5) – f(-9)?

Answer: g(5) – f(-9) = 26

Question 7/10

Suppose h(w) = 2w. What is h(v)?

Answer: h(v) = 2(v)

h(v) = 2v

Question 8/10

What does the function notation g(7) represent? (what is the input and output)

Answer: g(7) is the output, the input is 7

Question 9/10

Suppose g(x) = 3x + 2. Describe, in words, what the function g does.

Answer: The function g takes an input, multiplies by 3, and then adds 2.

Question 10/10

Write in function notation “the function g takes an input y adds 3, and then multiplies by 2.”

Answer: g(y) = 2(y + 3)

HW#6 – Page 11. f(4) = 4

2. g(2) = -4

3. g(-3) = 21

4. f(-5) = -14

5. f(3x) =6x – 4

6. f(g(-2)) = 20

7. g(t) = t2 – 4t

8. f(h) = 2h – 4

9. f(q + 1) = 2q – 2

10. f(2) + g(-2) = 12

11. g(-b) = b2 + 4b

12. f(r – 1) = 2r – 6

HW#6 – Page 113.

HW#6: Functions

Do Page 4 – counted as a quiz grade Use your notes/practice problems as you are

finishing the HW

SWBAT… find the domain and range of functions

Agenda 1. Review HW 6 – Page 1 – 3 (15 min)2. Domain & range (35 min)

Warm-Up:

HW#6: Functions – Page 5 – 6

Tues, 11/6

Real Numbers: All numbers on the number line. This includes positives and negatives, integers and rational numbers, square roots, cube roots , π, etc.

The domain of a function is the set of numbers that you can plug into the function and get out something that makes sense (think about how you get an “error” on your calculator.)

When finding the domain, remember:1. The denominator of a fraction cannot be zero

2. The values under a square root sign must be positive

Find the domain for the function

f(x) = x2 + 2. Explain. The domain is “all real numbers of x” because there are no restrictions on the

value of x.

The range of a function is the possible y values of a function that result when we substitute all the possible x-values into the function.

When finding the domain, remember:1. Substitute different x-values into the expression to see what is

happening for y.

2. Draw a sketch! In math, it's very true that a picture is worth a thousand words

Find the range for the function f(x) = x2 + 2. Explain.

The range is “y ≥ 2” since x2 + 2 is never less than 2.

1. What is the domain and range of f(x) = x2 + 3? Explain.

Domain: All real numbers because there are no restrictions on x.

Range: y ≥ 3 because x2 + 3 is never less than 3.

What is the domain of ?4 xy

4 xy What is the range of ?

What is the domain and range of ?

Domain: x ≥ 1

Range: y ≥ 0

1 xy

To use the symbols of algebra, we could write the domain as

xx :Does that look like a foreign

language?Let’s translate:

The curly braces just tell us we have a set of numbers.

The x reminds us that our set contains x-values.

x

The colon says, such that

:x

: xx

The symbol that looks like an e says, belongs to . . .

And the cursive, or script, R is short for the set of real numbers.

xx:

R, the set of real numbers.”

So we read it, “The setof x

:

such that x belongs to

x x

What is the domain of ?

0: xx

xy

10

“The set of x such that x does not equal 0.”

The domain would be _________

What is the domain of ?

31

x

y

The domain would be ___________ 3: xx

“The set of x such that x does not equal 3.”

What is the domain of ?xy

The domain would be __________ 0: xx

“The set of x such that x is greater than or equal to 0.”

What is the domain of y = -(x – 4)?

The domain would be __________ xx :

“The set of x such that x is all real numbers.”

Find the domain of each function:

51

.1

x

y

174.2 xy

9.3 xy

99.4 2 xy

Answers:

5:.1 xx

xx :.2

xx :.4

9:.3 xx

SWBAT… find the domain and range of functions

Agenda 1. Warm-Up (10 min)2. Review HW#5 – page 5-6 (10 min)

Warm-Up:1. What is the domain of f(x) = -3(4x − 5)?2. What is the domain of

HW#6: Study guide

Tues, 11/15

5

5

x

xy