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INSTRUCTORSCarleen EatonGrant Fraser
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Lecture Comments (5)

0 answers

Post by Munqiz Minhas on September 7 at 07:36:08 PM

You should include economic applications problems (profit functions)! I have a test on this and struggling a little.

1 answer

Last reply by: Dr Carleen Eaton
Thu Jul 31, 2014 6:58 PM

Post by Philippe Tremblay on July 17, 2014

At 23:40, the 'open circle' should be face to 3 (value of y).

0 answers

Post by Kavita Agrawal on May 20, 2013

Example 1 is a step function, not an absolute value function as you labelled in the side bar.

0 answers

Post by Tearion Scott on June 25, 2011

Why don't you include application of the problems related to economics or business? Still you do a great job. :-)

Special Functions

  • Step functions have graphs that are a series of steps. The steps can be going upward or downward.
  • Constant functions are of the form f(x) = c, where c is a real number. Their graphs are horizontal lines.
  • A piecewise function is defined differently on different intervals in the domain. The graph consists of a different graph on each interval
  • The absolute value function is an example of a piecewise function. Its graph looks like a V, translated horizontally or vertically.

Special Functions

Graph f(x) = [x]
  • Recall that this function is called the Greatest Integer Function. Another appropriate name for this function is the floor function because whatever value you get inside, it's brought to the floor, or to the number less than or equal to that whole integer.
  • Modern computer algebra systems have a built - in function g(x) = floor(x + 2) to allow you to work with these types of functions.
  • The greatest integer that is less than or equal to 4.7
  • Complete the table below, remember, you're looking for the floor of every number inside the [  ].
  • x f(x) = [x]
    -2.9  
    -2.1  
    -2  
    -1.9  
    -1.1  
    -1  
    0  
    0.9  
    1  
    1.1  
    1.9  
    2.1  
    2.9  
    3  
    3.1  
    3.9  
    4.0  
    4.1  
  • x f(x) = [x]
    -2.9 [-2.9]= floor(-2.9) = -3
    -2.1 [-2.1]= floor(-2.1) = -3
    -2 [-2]= floor(-2) = -2
    -1.9 [-1.9]= floor(-1.9) = -2
    -1.1 [-1.1]= floor(-1.1) = -2
    -1   [-1]= floor(-1) = -1
    0 [0]= floor(0) = 0
    0.9 [0.9]= floor(0.9) = 0
    1 [1]= floor(1) = 1
    1.1 [1.1]= floor(1.1) = 1
    1.9 [1.9]= floor(1.9) = 1
    2.1 [2.1]= floor(2.1) = 2
    2.9 [2.9]= floor(2.9) = 2
    3 [3]= floor(3) = 3
    3.1 [3.1]= floor(3.1) = 3
    3.9 [3.9]= floor(3..9) = 3
    4.0 [4.0]= floor(4) = 4
    4.1 [4.1]= floor(4.1) = 4
  • Graph it and follow the convention shown in lecture. This can't be shown in the graph provided
Graph f(x) = [x + 3]
  • Recall that this function is called the Greatest Integer Function. Another appropriate name
  • for this function is the floor function because whatever value you get inside, it's brought
  • to the floor, or to the number less than or equal to that whole integer. Modern computer algebra systems
  • have a built - in function g(x) = floor(x + 2) to allow you to work with these types of functions.
  • The greatest integer that is less than or equal to 4.7
  • Complete the table below, remember, you're looking for the floor of every number inside the [ ].
  • x f(x) = [x+3]
    -2.9  
    -2.1  
    -2  
    -1.9  
    -1.1  
    -1  
    0  
    0.9  
    1  
    1.1  
    1.9  
    2.1  
    2.9  
    3  
    3.1  
    3.9  
    4.0  
    4.1  
  • x f(x) = [x+3]
    -2.9 [-2.9+3]= floor(0.1) = 0
    -2.1 [-2.1+3]= floor(0.9) = 0
    -2 [-2+3]= floor(1) = 1
    -1.9 [-1.9+3]= floor(1.1) = 1
    -1.1 [-1.1+3]= floor(1.9) = 1
    -1 [-1+3]= floor(2) = 2
    0 [0+3]= floor(3) = 3
    0.9 [0.9+3]= floor(3.9) = 3
    1 [1+3]= floor(4)= 4
    1.1 [1.1+3]= floor(4.1) = 4
    1.9 [1.9+3]= floor(4.9) = 4
    2.1 [2.1+3]= floor(5.1) = 5
    2.9 [2.9+3]= floor(5.9) = 5
    3 [3+3]= floor(6) = 6
    3.1 [3.1+3]= floor(6.1) = 6
    3.9 [3.9+3]= floor(6.9) = 6
    4.0 [4.0+3]= floor(7) = 7
    4.1 [4.1+3]= floor(7.1) = 7
  • Graph it and follow the convention shown in lecture. This can't be shown in the graph provided
Graph f(x) = |x| + 1
  • This is the graph of the absolute value function of x.
  • Remember that the absolute valure refers to the distance away from zero, eg. | − 4| = 4,| − 6| = 6
  • Complete the table of values below and graph
  • x f(x) = |x| + 1
    -3  
    -2  
    -1  
    0  
    1  
    2  
    3  
    4  
  • x f(x) = |x| + 1
    -3 |−3| + 1 = 3 + 1 = 4
    -2 |−2| + 1 = 2 + 1 = 3
    -1 |−1| + 1 = 1 + 1 = 2
    0 |0| + 1 = 0 + 1 = 1
    1 |1| + 1 = 1 + 1 = 2
    2 |2| + 1 = 2 + 1 = 3
    3 |3| + 1 = 3 + 1 = 4
    4 |4| + 1 = 4 + 1 = 5
  • Graph it.
Graph f(x) = − |x| + 1
  • This is the graph of the absolute value function of x.
  • Remember that the absolute valure refers to the distance away from zero, eg. | − 4| = 4,| − 6| = 6
  • Complete the table of values below and graph
  • x f(x) = −|x| + 1
    -3  
    -2  
    -1  
    0  
    1  
    2  
    3  
    4  
  • x f(x) = −|x| + 1
    -3 −|−3| + 1 = − 3 + 1 = −2
    -2 −|−2| + 1 = −2 + 1 = −1
    -1 −|−1| + 1 = −1 + 1 = 0
    0 −|0| + 1 = −0 + 1 = 1
    1 −|1| + 1 = −1 + 1 = 0
    2 −|2| + 1 = −2 + 1 = −1
    3 −|3| + 1 = −3 + 1 = −2
    4 −|4| + 1 = −4 + 1 = −3
  • Graph it
Graph f(x) = |x + 3| + 1
  • This is the graph of the absolute value function of x.
  • Remember that the absolute valure refers to the distance away from zero, eg. | − 4| = 4,| − 6| = 6
  • Complete the table of values below and graph
  • x f(x) = |x+3| + 1
    -6 | − 6 + 3| + 1 = 3 + 1 = 4
    -5 | − 5 + 3| + 1 = 2 + 1 = 3
    -4 | − 4 + 3| + 1 = 1 + 1 = 2
    -3 | − 3 + 3| + 1 = 0 + 1 = 1
    -2 | − 2 + 3| + 1 = 1 + 1 = 2
    -1 | − 1 + 3| + 1 = 2 + 1 = 3
    0 |0 + 3| + 1 = 3 + 1 = 4
    1 |1 + 3| + 1 = 4 + 1 = 5
  • x f(x) = |x+3| + 1
    -6 | − 6 + 3| + 1 = 3 + 1 = 4
    -5 | − 5 + 3| + 1 = 2 + 1 = 3
    -4 | − 4 + 3| + 1 = 1 + 1 = 2
    -3 | − 3 + 3| + 1 = 0 + 1 = 1
    -2 | − 2 + 3| + 1 = 1 + 1 = 2
    -1 | − 1 + 3| + 1 = 2 + 1 = 3
    0 |0 + 3| + 1 = 3 + 1 = 4
    1 |1 + 3| + 1 = 4 + 1 = 5
  • Graph it.
Graph f(x) = |x − 3| + 1
  • This is the graph of the absolute value function of x.
  • Remember that the absolute valure refers to the distance away from zero, eg. | − 4| = 4,| − 6| = 6
  • Complete the table of values below and graph
  • x f(x) = |x−3| + 1
    -1
    0
    1
    2
    3
    4
    5
    6
  • x f(x) = |x−3| + 1
    -1 | − 1 − 3| + 1 = 4 + 1 = 5
    0 |0 − 3| + 1 = 3 + 1 = 4
    1 |1 − 3| + 1 = 2 + 1 = 3
    2 |2 − 3| + 1 = 1 + 1 = 2
    3 |3 − 3| + 1 = 0 + 1 = 1
    4 |4 − 3| + 1 = 1 + 1 = 2
    5 |5 − 3| + 1 = 2 + 1 = 3
    6 |6 − 3| + 1 = 3 + 1 = 4
  • Graph it.
Graph f(x) = x2  if  0 ≤ x ≤ 3x − 3  if  x < 0
  • There are two different pieces/sections in the graph, the function is defined differently for different intervals in the domain.
  • Whenever you see a > or < sign in the domain, you my still calculate the value, however, when graphed, it must be represented with an open circle.
  • Complete the table below and graph the Piece - wise Function.
  • Section 1:
    Domain: 0 ≤ x ≤ 3

    x f(x) = x2
    0*  
    1  
    2  
    3*  


    *must be closed circle
  • Section 1:
    Domain: 0 ≤ x ≤ 3

    x f(x) = x2
    0* (0)2 = 0
    1 (1)2 = 1
    2 ( 2)2 = 4
    3* ( 3 )2 = 9


    *must be closed circle
  • Section 2:
    Domain: x < 0

    x f(x) = x−3
    0*  
    1  
    2  
    3  


    *must be closed circle
  • Section 2:
    Domain: x < 0

    x f(x) = x-3
    0* 0-3=-3
    1 -1-3=-4
    2 -2-3=-5
    3 -3-3=-6


    *must be closed circle
  • Graph the piece - wise function. Pay close attention to open and close circles on the graph.
Graph f(x) = [3/4]x−1  if  4 ≤ x < 8 −[5/4]x+3  if  0 < x < 4
  • There are two different pieces/sections in the graph, the function is defined differently for different intervals in the domain.
  • Whenever you see a > or < sign in the domain, you my still calculate the value, however, when graphed, it must be represented with an open circle.
  • Complete the table below and graph the Piece - wise Function.
  • Notice how both sections are linear functions, therefore, you may find the end - points of the intervals of the domain only. Play close attention to open or closed circles.
  • Section 1:
    Domain: 4 ≤ x < 8

    x f(x) =[3/4]−1
    4 (Closed Circle)  
    8 (Open Circle)  
  • Section 1:
    Domain: 4 ≤ x < 8

    x f(x) =[3/4]−1
    4 (Closed Circle) [3/4](4)−1=3−1=2
    8 (Open Circle) [3/4](8)−1=6−1=5
  • Section 2:
    Domain: 0 < x < 4

    x f(x) =−[5/4]+3
    0 (Open Circle)  
    4 (Open Circle)  
  • Section 2:
    Domain: 0 < x < 4

    x f(x) =−[5/4]+3
    0 (Open Circle) −[5/4](0)+3=3
    4 (Open Circle) −[5/4](4)+3=−5+3=−2
Graph f(x) = x2−1  if  −3 ≤ x < 1 3x+1  if  1 < x ≤ 5
  • There are two different pieces/sections in the graph, the function is defined differently for different intervals in the domain.
  • Whenever you see a > or < sign in the domain, you my still calculate the value, however, when graphed, it must be represented with an open circle.
  • Complete the table below and graph the Piece - wise Function.
  • Notice how the second sections is a linear function, therefore, you may find the end - points of the interval of the domain only. Play close attention to open or closed circles.
  • Section 1
    Domain: −3 ≤ x < 1

    x f(x) =x2−1
    -3 (Closed Circle)  
    -2  
    -1  
    0  
    1 (Open Circle)  
  • Section 1
    Domain: −3 ≤ x < 1

    x f(x) =x2−1
    -3 (Closed Circle) (−3)2−1=9−1=8
    -2 (−2)2−1=4−1=3
    -1 (−1)2−1=1−1=0
    0 (0)2−1=0−1=−1
    1 (Open Circle) (1)2−1=1−1=0
  • Section 2
    Domain: 0 < x ≤ 5

    x f(x) = 3x+1
    1 (Open Circle)  
    5 (Closed Circle)  
  • Section 2
    Domain: 0 < x ≤ 5

    x f(x) = 3x+1
    1 (Open Circle) 3(1)+1=4
    5 (Closed Circle) 3(5)+1=16
  • Graph the piece - wise function. Pay close attention to open and close circles on the graph.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Special Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Step Functions 0:07
    • Example: Apple Prices
  • Absolute Value Function 4:55
    • Example: Absolute Value
  • Piecewise Functions 9:08
    • Example: Piecewise
  • Example 1: Absolute Value Function 14:00
  • Example 2: Absolute Value Function 20:39
  • Example 3: Piecewise Function 22:26
  • Example 4: Step Function 25:25

Transcription: Special Functions

Welcome to Educator.com.0000

In today's session, we are going to talk about several special functions.0002

The first one we are going to discuss is the step function; and the step function makes a unique-looking graph.0007

It is a function that is constant for different intervals of real numbers.0013

And the result is a graph that is a series of horizontal line segments, so they look like steps; and that is where the name "step function" comes from.0018

The best way to understand this is through an example.0028

So, for example, if apples were sold at a price of a dollar per pound, and the price is such that0030

you are charged $1 for each pound, or any part of a pound--in other words, they round up in order to determine the price;0053

if you had a pound and a half of apples, they are going to charge you $1 for the pound, and then another $1 for the half pound.0070

So, they charged you a dollar for a pound, and then any part of a pound is considered a full pound in terms of pricing.0079

So, if x is the number of pounds, and y is the cost, then let's see what kind of values we get and what the graph looks like.0088

OK, if I have .8 pounds, I am going to get charged $1: they are going to round up.0104

If I have a full pound, I am going to get charged $1; if I have a little bit over a pound--I have 1.2 pounds--0113

I am going to get charged $1 for the first pound, and then that .2 is going to be another dollar, so $2.0124

1.4 pounds--again, $2, and on up...2 pounds--a dollar for the first pound and a dollar for the second pound.0135

2.5--$2, and then the .5 is another $1, so that bumps me up to $3; until I hit 3 pounds, it is $3, also.0145

3.8: $1 for the first pound, $1 for the second pound, $1 for the third pound, and another $1 for that .8, so $4.0155

So, you can see that the function is constant for different intervals.0166

So, for this first interval, from a little bit above 0, all the way to 1, the y-value is constant; it is $1.0170

For the next interval, which is just above 1, all the way to 2, including 2, it is going to be $2.0180

Once I get above 2, up to and including 3, it is $3; and so on.0188

So, this is constant for different intervals of real numbers.0193

Looking at what the graph looks like: any part of a pound up to and including a dollar for that first pound is a dollar.0198

Now, 0 pounds of apples are going to be $0; so I am not going to include 0.0207

But even .1...the slightest part of a pound is going to be $1, so I put an open circle to indicate that 0 is not included;0213

but just above that is, all the way up to and including 1; since 1 is included (1 is also a dollar), I am going to put that as a closed circle.0224

So here, I have pounds; and here, I have the cost on the y-axis in dollars.0233

Now, once I get just above 1 (say 1.1 pounds), they are going to charge me $2--not including 1, but just above it--open circle.0243

All the way up to 2...at 2 pounds, I will also be charged $2.0254

Once I hit just above 2, I am going to be charged $3, all the way to 3 and including 3.0260

And on...so, you can see how this looks like a series of steps, and how this is a result of the fact that the function is constant for different intervals of x.0270

This function is the same for this entire interval; then it is the same for the second interval; and on.0287

So, this is a step function.0292

A second type of function that you will be working with is an absolute value function.0295

And these functions have special properties: looking first at f(x) = |x|, just the simplest case, here f(x) is just going to be the absolute value of x.0301

When x is 0, f(x) is also 0; when x is 1, the absolute value is 1; and so on for positive numbers.0317

Now, let's look at negative numbers: for -1, the absolute value is 1; -2--the absolute value of x is 2; and on.0329

The result is a certain shape of graph: when x is 0, f(x) is 0; x is 1, f(x) is 1; x is 2, f(x) is 2; and on.0343

Now, for negative numbers: -1, f(x) is 1; -2, f(x) is 2; -3, f(x) is 3; and it is going to continue on like that.0358

So, absolute value graphs are v-shaped; so we are going to end up with a v-shaped graph.0372

Depending on the function, the graph can be shifted up, or it can be shifted to the right or to the left; and let's see how that could happen.0385

Let's now let f(x) equal (here f(x) equals |x|)...let's say f(x) equals the absolute value of x, plus 1.0393

OK, so we are given x, and the absolute value of x is 0; we are adding 1 to them, so this is going to become 1.0404

The absolute value of x is 1; 1 + 1 is 2; 3; 4; the absolute value of -1 is 1; 1 + 1 is 2.0412

The absolute value of -2 is 2; add one to that--it is 3; and add 1 to 3 to get 4.0425

OK, so it is the same as this, except increased by 1: each value of the function has been increased by 1.0432

So now, let's see what my graph is going to look like.0439

Right here, I have the graph for f(x) = |x|; now, I am going to look at this graph.0443

When x is 0, f(x) is 1; when x is 1, f(x) is 2; when x is 2, f(x) is 3; so you can see what is happening.0451

And then here, I have x is 3, f(x) is 4; negative values--when x is -1, f(x) is 2; when x is -2, f(x) is 3; when x is -3, f(x) is 4.0465

OK, I am drawing the line through this: this is the graph of f(x) = |x| + 1.0492

So you see that this graph, the v-shaped graph, is simply shifted up by 1.0509

And again, you can also shift this from side to side; and we will see an example of that later on.0513

So, in the absolute value function, it is very important to find both negative and positive values of the function.0519

So, assign x 0; assign it some positive values; and it is very important to find what f(x) will be when x is negative,0526

because if I didn't--if I picked only positive values--I would end up with half of a graph.0537

So, to get the entire v-shape, choose negative and positive values for x.0541

The third special function that we are going to discuss is called a piecewise function.0548

And a piecewise function is a function that is described using two or more different expressions.0553

The result is a graph that consists of two or more pieces.0560

Just starting out with one that consists of two pieces: the notation is usually like this--one large brace on the left:0564

f(x) equals x + 2 for values of x that are less than 3; and f(x) equals 2x - 3 for values of x that are greater than or equal to 3.0573

So you see that, for different intervals of the domain, the function is defined differently.0590

So, it is a function that is described using two or more different expressions.0597

So, for the part of the domain where x is less than 3, this is the function.0600

For the interval of the domain that is greater than or equal to 3, this is the function that I am going to use.0605

Let's see what happens: let's first use this part of the function where f(x), or y, equals x + 2.0613

And for the domain, remember that x is going to be less than 3.0625

So, I will go ahead and start out with 2: when x is 2, f(x) (or y) is 4.0630

When x is 1 (I have to remain at x-values less than 3), f(x) is 3; when x is 0, f(x) is 2.0636

Just picking a negative number: when x is -4, f(x) is -2.0648

I am going to graph that here; OK, when x is 2, f(x), or y, is 4; x is 1, y is 3; x is 0, y is 2; x is -4, y is -2.0656

OK, now, this is for values of x that are less than 3; 3 is about here.0689

Therefore, anything just below 3, but not including 3, is going to be part of this graph.0698

So, we are going to use an open circle here to indicate that 3 is not going to be included as part of this function--the domain of this function.0703

So, it is going to begin at just below 3 and continue on indefinitely; that is the first piece of the graph.0716

The second piece of the graph is for x such that x is greater than or equal to 3; and here, y is going to be 2x - 3.0723

So, it is greater than or equal to 3, so I am first going to let x equal 3.0735

3 times 2 is 6, minus 3 is 3; getting larger--when x is 4, it is 2 times 4 is 8, minus 3 is 5; when x is 5, 5 times 2 is 10, minus 3 is 7.0738

Now, this does include 3--this section of the graph--this piece; so, when x is 3, y is 3, right here.0755

When x is 4, y is 5, right here; when x is 5 (let me shift that over just a bit), y is 7; OK.0768

Now, if you look, this actually did end up including all possible values of x (all real numbers),0801

because when x is less than 3, I use this function; and then, as soon as x becomes 3 or greater, I shift to this other function.0807

So, you can see how there are two pieces to the graph; and you actually can have situations where there are more than 2.0816

You could be given, say, f(x) is 4x + 3, 2x + 7, and x - 1, and then given limits on the domain for each of those.0822

So, there are at least two pieces; however, there can be more.0836

In Example 1, we have a greatest integer function.0840

Before we start working in this, let's just review what we mean by the greatest integer function.0845

So, when you see this notation with the brackets, let's say that you have a number in here, such as 4.7.0848

What this is saying is that this value is equal to the greatest integer that is less than or equal to 4.7; so that is 4.0855

It is the greatest integer less than or equal to whatever is in here; if it was 2.8, it would be 2.0866

Be careful with negative numbers: let's say I have -3.2--the temptation is to say, "Oh, that is equal to -3";0875

but if you look at it on the number line, -3.2 is right about here; OK, so if I have -3.2,0882

and I am trying to find the greatest integer that is less than or equal to 3.2, it is going to have to be something over here--smaller.0893

So, it is actually going to be -4; so just be careful when you are working with negative numbers.0903

Whatever is inside here--whatever that value is--the function is equal to the greatest integer that is less than or equal to this value in here.0907

Understanding that, you can then find the graph; so let's find a bunch of points for this,0920

so we make sure we know what is going to happen with various situations.0927

When x is 0, this inside here is going to be 2; the greatest integer less than or equal to 2 is 2.0935

When x is .6, then you are going to get 2.6 in here; the greatest integer less than or equal to 2.6 is 2.0945

.8--I get 2.8; again, I round down to 2.0953

All right, so when I hit 1, 1 plus 2 is 3, and the greatest integer less than or equal to 3 is 3.0958

Slightly above 1: that is going to give me 1.2 + 2 is 3.2; the greatest integer less than or equal to 3.2 is also 3.0967

OK, so you can get the idea of what this is going to look like.0976

And that continues on; and then, when we hit 2, 2 + 2 is 4; the greatest integer is going to be 4.0980

For negative numbers: let's take -.5: -.5 and 2 is 1.5; the greatest integer less than or equal to 1.5 is going to be 1.0988

Now, notice: I have a negative number for x, but this did not come out to be a negative number; so that is different from the case I was discussing there.1005

Let's go a little bit bigger--let's say -3: -3 and 2 is -1, and that is going to be -1.1014

Let's say I take -3.5: -3.5 and 2 is going to equal -1.5: again, just thinking about that to make sure you have it straight,1024

I have -1.5; so I have 0; I have -1; I have -1.5; I have -2; the greatest integer less than or equal to this is actually -2.1037

OK, now plotting this out: when x is 0, f(x) is 2; when x is slightly above 0 (it's .6), f(x) is 2; .8--it is 2, all the way up until I hit 1.1053

At 1, f(x) becomes 3; therefore, 1 is not included in this interval.1075

So, you can already see that this is going to be a step function, because we have intervals.1082

For different intervals of the domain, we have that same value for the range.1089

All right, for values between 1 and 2, f(x) will be 3; once we hit 2, I have to do an open circle, because at 2, the value for f(x) jumps up to 4.1095

OK, so you can see what this is going to look like; and that pattern is just going to continue.1112

Let's look over here at negative numbers: when x is slightly less than 0, then you are going to end up with an f(x) that is 1.1116

So, for values slightly less than 0, but not including 0, this is what you are going to end up with.1137

OK, looking, say, when x is -3: when x is -3, f(x) will be -1.1145

But when we go slightly more negative than that, when x is -3.5, f(x) is going to be -2; it is going to be down here.1161

So, the steps on this side are going to have the open circle on the right.1175

And I am going to jump down, and it is not going to include -2, because -2 and 2 is 0;1184

so -2 is going to be right here for the x-value, and the f(x) will be 0.1193

But as soon as I get to a little bit bigger than -2, the greatest integer is going to be down here.1199

OK, and so, we continue on like that with the steps; and you can see how this is a step function.1207

You just have to be very careful and pick multiple points until you can see the pattern1220

where for a certain interval of the domain, the range is a particular value.1225

OK, so that was a step function, and it involved the greatest integer function.1234

Example 2: now we are working with absolute value.1240

g(x) equals the absolute value of x, minus 3.1244

And we already know that the shape of this graph is going to be in a v.1248

But we don't know exactly where that v is going to land, so let's plot it out.1252

When x is 0, the absolute value of x is 0; minus 3--that gives me -3.1258

When x is 1, the absolute value is 1; minus 3...g(x) is -2.1264

When x is 2, the absolute value is 2; minus 3 is going to give me -1.1269

Now, let's pick some negative numbers for x, because that is really important to do with an absolute value graph.1277

When x is -1, the absolute value is 1, minus 3 gives me -2; you can already see that my v shape is going to occur.1283

When x is -2, the absolute value is 2; minus 3 is -1.1291

The absolute value of -3 is 3; minus 3 is 0; so this is enough to go ahead and plot.1299

x is 0; g(x) is -3; x is 1, g(x) is -2; x is 2, g(x) is -1; over here with the negative values,1305

when x is -1, g(x) is -2; when x is -2, g(x) is -3; when x is -3, g(x) is 0.1316

So, you can see that I have a v-shaped graph, and compared with my graph that would look like this,1325

that would have the v starting right here, it has actually shifted down by 3; that is an absolute value function.1338

Here you can see that you are given a piecewise function, because there are two different pieces.1348

And this could also be written in this notation.1357

There are two different sections to the graph; and we see that the function is defined differently for different intervals of the domain.1360

Starting with if x is greater than 2 (this is going to be for x-values where x is greater than 2): f(x) is going to be x + 1.1371

When x is 3, f(x) is 4; when x is 4, f(x) is 5; when x is 5, f(x) is 6; OK.1386

When x is 3, f(x) is 4; when x is 4, f(x) is 5; and it is going to go on up.1404

And that is going to go all the way, until just greater than 2.1416

2 is not going to be included on this graph, because it is a strict inequality; so I am going to use an open circle, and this is going to continue on.1422

Now, for x less than or equal to 2, I have a different situation: I am looking at f(x) is -2x.1430

OK, so when x is 2, 2 time -2 is -4; when x is 1, 1 times -2 is -2; when x is 0, f(x) is 0; when x is -2, that is -2 times -2, which is positive 4.1443

So, starting with x is 2: when x is 2, f(x) is -4; and that is including the 2.1464

When x is 1 (these are values less than or equal to 2, so I am getting smaller), f(x) is -2.1476

x is 0; f(x) is also 0; when x is -2, f(x) is up here at 4; OK, so I have a steep line going right through here.1494

So, you can see: this is a piecewise function consisting of two pieces; and here, one picks up where the other leaves off.1509

For values greater than 2, this is my graph; for values of x less than or equal to 2, this is my graph; so this is a piecewise function.1515

OK, this time, in Example 4, we have both greatest integer and absolute value in this function.1525

Recall that, for the greatest integer function, what that is saying is that whatever is inside this bracket--let's say it's 1.2--1532

it is asking for the greatest integer less than or equal to 1.2; in that case, this would be 1.1540

Or if I had 4.8, it would be 4.1545

For negative numbers, like -3.2, the greatest integer less than or equal to -3.2 is -4.1551

OK, now, since this is a bit complicated, it is helpful just to take it in stages.1560

So, I am going to look first at what the greatest integer of x is; and then, I am going to look for the absolute value of what that is.1567

If x is .2, the greatest integer less than or equal to .2 is 0; the absolute value of 0 is 0; so this is the function that we are looking for.1578

And the same would hold true of .5: round down to 0; the absolute value is 0.1590

When we hit 1, the greatest integer less than or equal to 1 is 1, and the absolute value of that is 1.1596

1.2: again, we are going to go down to the greatest integer that is less than or equal to 1.2, which is 1; and the absolute value is 1.1608

The same for 1.8, and all the way up until 2; once we hit 2, the greatest integer less than or equal to 2 is 2; the absolute value is 2.1621

So, that is working with positive numbers, greatest integer, and the absolute value; it is the same; OK.1629

So, let's go to negative: for -.4, the greatest integer that is less than or equal to -.4...I am looking, and I have 0, and 1,1636

and -1, and -.4 is about here; so I am going to go down to -1; the absolute value of that is 1.1650

Here you can see that the greatest integer is not the same as the absolute value.1659

Or for -1, the greatest integer less than or equal to -1 is -1; the absolute value is 1.1664

-1.8: the greatest integer that is less than or equal to -1.8...I am going to go down to -2; and the absolute value is 2.1674

For -2, the greatest integer less than or equal to -2 is -2; the absolute value is 2.1686

So, you see that there are intervals here--intervals of the domain end up with the same value for the function.1694

So, I am going to have a step function.1704

But remember that absolute value graphs are v-shaped, so I am going to end up with a v-shaped step function.1709

Let's plot these out: for 0, the greatest integer of 0 would be 0, and then the absolute value would be 0.1715

So, with 0, we are going to include it; and for all values up to but not including 1, the function is going to equal 0.1726

Once we get to 1, I have an open circle, because it is not included.1736

When x is 1, f(x) is 1; so I am going to jump up here.1740

All the values between 1 and 2, but not including 2, will have an f(x), or a y-value, that is 1.1746

As soon as I hit 2, open circle: I am going to jump up, and once I hit 2, f(x) is 2, all the way up to, but not including, 3.1754

And it is going to go on that way: and you see now, we have the step function, and it is v-shaped like absolute value.1771

Let's look over here on the negative side of things.1778

For -.4, somewhere in here, it is going to equal 1; -1 is also equal to 1; so here, on the left side, I have a closed circle, and an open circle on the right.1782

It is the opposite of what I had over here.1799

When I get to less than -1, my value for f(x) is going to jump up to 2; this is a closed circle;1802

I get slightly less than, but not including, -1; it is going to jump up to 2.1815

-2: my value is also 2, and everything in between; and then, when I get to just slightly more negative than -2, like -2.1, it is going to jump up to 3.1822

You can see how this is v-shaped, and it is a step function.1841

The step function comes from it being the greatest integer function; the v shape comes from that absolute value.1847

And you also just had to be careful how you are doing the open and the closed circles; OK.1853