module 1 lesson 1 remediation notes

Remediation Notes

Relation Function

Every equation/graph/set of ordered pairs represents a relation, but sometimes a relation is a function.

Functions are just relations in which the x values of its points (ordered pairs) do not repeat.

If a graph passes the vertical line test, then it is the graph of a function.

To determine if a graph is a function, we use the vertical line test.

If it passes the vertical line test then it is a function.

If it does not pass the vertical line test then it is not a function.

Vertical Line Test:

1.Draw a vertical line through the graph.

2. See how many times the vertical line intersects the graph at any one location.

If Only Once – Pass (function)

If More than Once – Fail (not function)

Is this graph a function?

Yes, this is a function because it passes the vertical line test.

Only crosses at one point.

Is this graph a function?

No, this is not a function because it does not pass the vertical line test.

Crosses at more than one point.

To determine if a table represents a function, we look at the x columnx column

(domain).

If each number in the x column appears only onceonce in that column, it

is a function.

You can use the vertical line test to determine whether a relation is a function.

Vertical Line Test

If no vertical line intersects agraph in more than one point,

the graph represents a function.

If some vertical line intercepts agraph in two or more points, the

graph does not represent a function.

Relations and Functions Relations and Functions

y

x

(-4,3) (2,3)

(-1,-2)

(0,-4)

(3,-3)

State the domain and range of the relation shownin the graph. Is the relation a function?

The relation is:

{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }

The domain is:

{ -4, -1, 0, 2, 3 }

The range is:

{ -4, -3, -2, 3 }

Each member of the domain is paired with exactly one member of the range,so this relation is a function.

Relations and Functions Relations and Functions

Is this relation a function?

Yes, this is a function because each number in the x column only appears once.

Every number just appears once.

X Y

1 5

2 6

3 5

4 8

Is this relation a function?

No, this is not a function because 10 appears in the x column more than once.

X Y

24 7

6 9

10 8

10 10

The number 10 appears more than

once.

To Evaluate a Function for f(#):Plug the # given in the (#) into all x’s

Simplify

Try these…http://www.mathslideshow.com/Alg2/Lesson2-1/fv4.htm

Functions

Remember f(x), g(x), h(x), … all just mean y.

We use f(x), g(x), h(x), … when we have more than one y = equation.

ReviewEvaluate for

Evaluate for

52)( 2 xxxf

).1(f825)( 3 xxxf

).3(f

f(3) = (3)2 – 2(3) + 5f(3) = 8

f(-1) = 5(-1)3 – 2(-1) – 8f(-1) = -11

©1999 by Design Science, Inc. 15

Basic function operationsSum

Difference

Product

Quotient

, 0f xf

f g x x g xg g x

f g x f x g x+ +

– –f g x f x g x

)( )( xgxfxgf

32)( xxf 95)( xxg

127)()(

9532)()(

)95()32()()(

xxgxf

xxxgxf

xxxgxf

You MUST DISTRIBUTE the NEGATIVE

32)( xxf 95)( xxg

273310)()(

27151810)()(

9532)()(

2

2

xxxgxf

xxxxgxf

xxxgxf

You MUST FOIL

If you are given a set of ordered pairs or a graph (which you would find the ordered pairs all by yourself)The x values are the DOMAINThe y values are the RANGE

Domain and Range:

{ (-3,5) , (-1, 6), (0, 4), (2, 3.5), (6, 13), (6, 29}

Range: { 3.5, 4, 5, 6, 13, 29}Domain: { -3, -1, 0, 2, 6 }

Domain and Range:If the equation is a line (y = mx + b or y = #)

DOMAIN AND RANGE ARE ALL REAL NUMBERS

ALWAYS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

If there is an x in the denominator of a fraction, you need to find the value of x that makes the ENTIRE DENOMINATOR equal zero.

This number is the EXCEPTION to the DOMAIN of all real numbers.

Domain and Range:

x

anythingbecould Domain is all real numbers except 0

5

xanythingbecould

Domain is all real numbers except 5

9

xanythingbecould

Domain is all real numbers except -9

If you are given a line segmentThe DOMAIN (x values) is written like

# < x < #The RANGE (y values) is written like # < y < #

# < x < ## < y < #

Domain and Range:

If you are given a parabolaThe DOMAIN is ALWAYS ALL REAL

NUMBERSThe RANGE (y values) is written like y > # or

y < #

Domain and Range:

Find domain and range from an equation

Most of the functions you study in this course will have all real numbers for both the domain and range. We’ll only look at the domain for exceptions:

1. Fractions: cannot have the denominator (bottom) = 0, so domain cannot be any x-value that makes the denominator= 0

Examples

Domain: x≠0 Domain: x≠3 (it’s okay for x=0 on top!)

Domain: x≠1 or -1 because they both make the denominator=0Question: How can you calculate which values make the denominator = 0? Set up

the equation denominator = 0 and solve it. Those values are NOT allowed!

xxf

3)( 3

)(

x

xxf

1

12

2

x

xy

Review

12 x 34 x

34 yDomain: {-3,-2,1,3} Domain:

Domain: {x| }

Range: {0, -3} Range: y=4 or {4} Range: {y| }

*Don’t repeat y *x is between -2 and 1 *This is “set notation”

y

x

5

5

-5

-5

● ● ● ●

y

x

5

5

-5

-5

y

x

5

5

-5

-5

3x

0y2y

Domain: Domain: x is any real # Domain: x is any real #Range: Range: Range: y is any real #*Graph continues rt *Graph continues down *Graph continues all ways

Examples

x

y

x

y

Does the graph represent a function? Name the domain and range.

YesD: all realsR: all reals

YesD: all realsR: y ≥ -6

x

y

x

y

Does the graph represent a function? Name the domain and range.

NoD: x ≥ 1/2R: all reals

NoD: all realsR: all reals

Visit these sites for remediation:

http://www.purplemath.com/modules/fcnops.htm

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut30b_operations.htm

http://teachers.henrico.k12.va.us/math/hcpsalgebra2/2-1.htm