review 1.1-1.3. a relation is a set of ordered pairs. {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} this is...

REVIEW 1.1-1.3

A relation is a set of ordered pairs.

{(2,3), (-1,5), (4,-2), (9,9), (0,-6)}This is a relation

The domain is the set of all x values in the relation

{(2,3), (-1,5), (4,-2), (9,9), (0,-6)}

The range is the set of all y values in the relation

{(2,3), (-1,5), (4,-2), (9,9), (0,-6)}

domain = {-1,0,2,4,9}

These are the x values written in a set from smallest to largest

range = {-6,-2,3,5,9}

These are the y values written in a set from smallest to largest

Domain (set of all x’s) Range (set of all y’s)

1

2

3

4

5

2

10

8

6

4

A relation assigns the x’s with y’s

This relation can be written {(1,6), (2,2), (3,4), (4,8), (5,10)}

A function f from set A to set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B.

Whew! What did that say?

Set A is the domain

123

4

5

Set B is the range

2

10864


Must use all the x’s


The x value can only be assigned to one y

This is a function ---it meets our

conditions

All x’s a

re

assigned

No x has more

than one y assigned

Set A is the domain

123

4

5

Set B is the range

2

10864

Must use all the x’s

Let’s look at another relation and decide if it is a function.

The x value can only be assigned to one y

This is a function ---it meets our

conditions

All x’s a

re

assigned

No x has more

than one y assigned

The second condition says each x can have only one y, but it CAN be the same y as another x gets assigned to.

123

4

5

2

10864

Is the relation shown above a function? NO Why not???

2 was assigned both 4 and 10

Evaluating Functions

632 2 xxxf

So we have a function called f that has the variable x in it.

Using function notation we could then ask the following:

Find f (2).

This means to find the function f and instead of having an x in it, put a 2 in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a 2.

623222 2 f

8668623422 f

Don’t forget order of operations---powers, then multiplication, finally addition & subtraction

Remember---this tells you what is on the right hand side---it is not something you work. It says that the right hand side is the function f and it has x in it.

632 2 xxxfFind f (-2).

This means to find the function f and instead of having an x in it, put a -2 in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a -2.

623222 2 f

20668623422 f


632 2 xxxfFind f (k).

This means to find the function f and instead of having an x in it, put a k in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a k.

632 2 kkkf

632632 22 kkkkkf


632 2 xxxfFind f (2k).

This means to find the function f and instead of having an x in it, put a 2k in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a 2k.

623222 2 kkkf

668623422 22 kkkkkf


xxxg 22

Let's try a new function

11211 2 g

Find g(1)+ g(-4).

248164244 2 g

2324141 So gg

Find the domain for the following functions:

12 xxf

Since no matter what value you choose for x, you won't be dividing by zero or square rooting a negative number, you can use anything you want so we say the answer is: All real numbers x.

2

3

x

xxg

If you choose x = 2, the denominator will be 2 – 2 = 0 which is illegal because you can't divide by zero. The answer then is: All real numbers x such that x ≠ 2.

means does not equalillegal if this

is zero

Note: There is nothing wrong with

the top = 0 just means the fraction = 0

Let's find the domain of another one:

4 xxh

We have to be careful what x's we use so that the second "illegal" of square rooting a negative doesn't happen. This means the "stuff" under the square root must be greater than or equal to zero (maths way of saying "not negative").

Can't be negative so must be ≥ 0

04 x solve this 4x

So the answer is:

All real numbers x such that x ≠ 4

Name all values of x that are not in the domain of the given function.

Given that x is an integer, state the relation representing each of the following by listing a set of ordered pairs. Then state whether the relation is a function or not.

And

{(0, -7), (1, -2), (2, 3), (3, 8)}

IS THIS A FUNCTION??? YES


And

{(-1, 3), (0, 0), (1, 3), (2, 24)}



And

{(-8, 4), (-7, 3), (-6, 2), (-5, 1), (-4, 0), (-3, 1)}


http://www.bssh.ac.uk/images/surgery.jpg

The sum f + g

xgxfxgf This just says that to find the sum of two functions, add them together. You should simplify by finding like terms.

1432 32 xxgxxf

1432 32 xxgf

424 23 xx

Combine like terms & put in descending order

The difference f - g

xgxfxgf To find the difference between two functions, subtract the first from the second. CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms.

1432 32 xxgxxf

1432 32 xxgf

1432 32 xx

Distribute negative

224 23 xx

The product f • g

xgxfxgf To find the product of two functions, put parenthesis around them and multiply each term from the first function to each term of the second function.

1432 32 xxgxxf

1432 32 xxgf

31228 325 xxx

FOIL

Good idea to put in descending order but not required.

The quotient f /g

xg

xfx

g

f

To find the quotient of two functions, put the first one over the second.

1432 32 xxgxxf

14

323

2

x

x

g

f Nothing more you could do here. (If you can reduce these you should).

The Composition Function

xgfxgf This is read “f composition g” and means to copy the f function down but where ever you see an x, substitute in the g function.

1432 32 xxgxxf

314223 xgf

51632321632 3636 xxxx

FOIL first and then distribute the 2

xfgxfg This is read “g composition f” and means to copy the g function down but where ever you see an x, substitute in the f function.

1432 32 xxgxxf

132432 xfg

You could multiply this out but since it’s to the 3rd power we won’t

Graphically, the x and y values of a Graphically, the x and y values of a point are switched.point are switched.

The point (4, 7)The point (4, 7)

has an inverse point has an inverse point of (7, 4)of (7, 4)

ANDAND

The point (-5, 3)The point (-5, 3)

has an inverse has an inverse point of (3, -5)point of (3, -5)

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Graphically, the x and y values of a point are switched.Graphically, the x and y values of a point are switched.

If the function y = g(x) If the function y = g(x) contains the pointscontains the points

then its inverse, y = gthen its inverse, y = g-1-1(x), (x), contains the pointscontains the points

xx 00 11 22 33 44

yy 11 22 44 88 1616

xx 11 22 44 88 1616

yy 00 11 22 33 44

Where is there a Where is there a line of reflection?line of reflection?

The graph of a The graph of a function and its function and its

inverse are inverse are mirror images mirror images about the line about the line

y = xy = xy = f(x)y = f(x)

y = fy = f-1-1(x)(x)

y = xy = x

Find the inverse of a function :Find the inverse of a function :

Example 1: Example 1: y = 6x - 12y = 6x - 12

Step 1: Switch x and y:Step 1: Switch x and y: x = 6y - 12x = 6y - 12

Step 2: Solve for y:Step 2: Solve for y:

Example 2:Example 2:

Given the function : Given the function : y = 3xy = 3x22 + 2 + 2 find the inverse: find the inverse:

Step 1: Switch x and y:Step 1: Switch x and y: x = 3yx = 3y22 + 2 + 2

Step 2: Solve for y:Step 2: Solve for y:

Ex: Find an inverse of y = -3x+6.• Steps: -switch x & y

-solve for y

y = -3x+6

x = -3y+6

x-6 = -3y

yx

3

6

23

1

xy

Find the zero of each function. Then graph the function.

1.) f(x) = 3x - 8 2.) f(x) = 19

Graphing Linear Equations and Inequalities

Graph 7x + y > –14

x

y

• Pick a point not on the graph: (0,0)

• Graph 7x + y = –14 as a dashed line.

• Test it in the original inequality.

7(0) + 0 > –14, 0 > –14

• True, so shade the side containing (0,0).

(0, 0)

Linear Equations in Two Variables

Example

Graph 3x + 5y –2

x

y

• Pick a point not on the graph: (0,0), but just barely

• Graph 3x + 5y = –2 as a solid line.


3(0) + 5(0) > –2, 0 > –2

• False, so shade the side that does not contain (0,0).

(0, 0)

Example


Graph 3x < 15

x

y

• Pick a point not on the graph: (0,0)

• Graph 3x = 15 as a dashed line.


3(0) < 15, 0 < 15

• True, so shade the side containing (0,0).

(0, 0)

Example


Graph this:

3x 5y 15

review 1.1-1.3. a relation is a set of ordered pairs. {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} this is...

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