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- Jonathan E Lopez on Reading Assignment: pp. 325-top 329
- Jonathan E Lopez on Reading Assignment: pp. 325-top 329
- Jonathan E Lopez on Reading Assignment: pp. 325-top 329
- Jonathan E Lopez on Reading Assignment: pp. 325-top 329
- Jonathan E Lopez on Reading Assignment: pp. 325-top 329

After reading this section I realized that there are a lot of things that I never realized applied to calculus such as the force of a dam or how to make a roller coaster ride smoothly. On pages 10-12 the book discussed the ways to represent a function. I don’t have a firm grasp on functions, domains, and range. But I did learn that the function has to do with dependent and independent variables, for one variable to be solved you must have one that is constant. I like the idea of the “machine” mentioned on page 11 where is like we are constantly putting in new variables and the machine spits out the answer once we apply the necessary steps. On pages 16-17, piecewise defined functions are discussed which I have a hard time understanding. I believe it has to do with functions and limitations applied on the domains of a particular function.

Reading these pages refreshed my memory on some of the basic calculus that I learned in high school. I never had trouble with any of the functions or the graph of functions but I did have difficulty understanding the piece-wise functions. Even after I re-read the section on piece-wise functions I still feel confused.

Hopefully the tax bracket example from class today helped. A piecewise function is basically made up of pieces of other functions. The tax bracket function, for example, is essentially made up of pieces of three lines. Which piece of the line we use to determine the amount of tax depends of x, the income.

Yes, you can think of restricting the domain of a function so that you have just a “piece” of its graph. If you do this several times so that you have several “pieces”, you can put these together to form a piecewise function. In the tax bracket example we looked at today, we basically took pieces of three lines, and put them together to form the tax function T(x).

I took AP Calculus AB in high school, so this chapter was pretty basic to me. It seems more like basic algebra rather than calculus. I think the description of the domain as all possible inputs and the range as all possible outputs was simple yet effective. Piecewise functions, in my opinion, are easiest to read once graphed and I generally think of them as separate functions while graphing them to avoid any confusion between the multiple formulas.

I took college calculus as a junior in high school, but have only taken statistics courses since then. This chapter was very beneficial as a refresher for some basic algebra that I had forgotten about. Specifically, the sections on the (f(a+h) – f(a))/h expression and the piecewise function were very helpful. Also, this section helped in refreshing that domain is input and range is output . If we could go over a couple example problems on the (f(a+h) – f(a))/h expression in class, I would find it very helpful. The piecewise function caused me a bit of trouble when I learned about it in high school, but after reading this section a couple of times, I feel that I have obtained a good grasp of this topic.

This section of reading was very helpful with refreshing old material within algebra and calculus from high school. I especially felt this with the in depth description of what a function is, the name of certain diagrams such as the arrow diagram, and the four types of ways to represent a function: verbally, numerically, visually, and algebraically.

For the example on page 16, why would you write an absolute value function as piecewise? Do you have to or is that just another way of writing it? It seems redundant and less simple than y=abs(x).

Writing the absolute value function as a piecewise function isn’t really necessary if you know what the absolute value function does (but the “definition” of the absolute value function is the piecewise description). The main point is this: when the way that you determine the output is dependent on the value of the input, you have a piecewise function. The absolute value function is an example of this, because you treat negative inputs and non-negative inputs differently.

I don’t like the way the book describes some of the basic concepts. The chapter was a good review of basic algebra, but some of the examples were written weird and explained in a way that seemed to make basic concepts a little tougher. The example 3 on page 12 lost me a little, I couldn’t quite see where they got (2a^2-5a+1). The other concepts seemed fine. I was fine with the functions, foil and the line test. Just like finite and AFM on page 14 not a fan of domain and range still.

The expression you mention, 2a^2-5a+1 is the value of f(a).

Notice that the numerator of what you are asked to evaluate is f(a+h)-f(a).

They calculate f(a+h) separately in lines 2-4 of the solution. From this, they must then subtract f(a), which is the value obtained by substituting a into the function in place of x. So f(a)=2a^2-5a+1.

I have taken calculus last year and have learned domain and range. After learning it already, reading the textbook seems to make it a bit harder than I remember. Example 2 (a&b) were very easy to understand because they are basic graphs. Example 3 reminded me of the step I always struggled with, filling in the a+h for the variables. The book goes too much into detail. I do not remember this -> {(x,f(x))/xED} The fist part is fine but what is the second part? I know it is domain but the “E” is odd. I also don’t have the actual symbol.

That funny-looking E you mention is a symbol that means “is in”. So “xED” really means “x is in D”. In other words, they are saying that if x “is in” D (the domain), then the point (x,f(x)) is on the graph of the function f.

This section is pretty basic. I liked how it gave the example of a machine to visualize what a function does. For example 3 on page 12, it doesn’t really explain substituting (a+h). All you have to do is plug in whatever is inside the paranthesis for x. Wherever there’s an x, you plug in (a+h). Thats it. But the book just shows the steps, it doesnt explain it, so it looks much more confusing that what it actually is. As for the piecewise functions, I think the book explains it pretty straightforward. No confusion there. Overall, this chapter is just algebra review.

I found that this section on functions was jus an OK refresher on the material. I really liked when the book would help break down the definition of a function and gave examples of easier ways to think of it, especially the machine analogy on page 11. I did, however, find the examples to be a bit hard to follow, especially example 3 on page 12. I was confused on why they showed the work they only showed the work for the (a+h) replacement and not the (a) replacement as well. Other than that it was ok, the piecewise functions were well explained however I think I may have to review some of the graphs on pages 16 and 17.

Generally calculating the first part of the numerator, f(a+h), is harder. So that is why they evaluate that part separately. The second part of the numerator, f(a), is generally a bit easier to determine because you simply replace the “x” in the function with an “a”. So since f(x)=2x^2-5x+1, then f(a)=2a^2-5a+1.

I took Calculus in high school so I’m already familiar with most of these concepts. But I thought that something on a graph was only a function if it could pass the vertical line test, so this is the only thing I’m slightly confused About đź¤”

The Vertical Line Test can be used to determine when a graph represents a function of x. If no vertical line intersects the graph more than once, we say the graph “passes the VLT”, and so represents a function of x.

If any vertical line intersects the graph more than once, we say the graph “fails the VLT”, and so does NOT represent a function of x.

In Figure 13 on page 15, for example, the left graph passes the VLT and the right graph fails the VLT.

I’ve taken a few years of algebra, calculus, and statistics; so this section was very easy and covered what we did in class, expanding only slightly with the absolute value functions on page 16. I was always taught to view functions as machines with inputs and outputs which is probably why it has stuck with me. The layout of the text vs. examples could be better to make it easier to follow, but that can’t be helped.

I have a good background in calculus as I took it in high school. It was a nice refresher to see the graphs that represent all of the functions. I think visual representation like that is one of the best ways to learn and make things clear. Seeing the graph for step functions helped me to understand them.

I never took calculus but I took pre-calc and

Reading this reminded me of my pre-calc textbook I read in high school. Pages 10-12 refreshed my memory on functions, graphs and evaluating algebratic equations. The terms were review to me and it was good that we went over it in class today and actually had to re-read it ourselves. Pages 16-17 was ok but I was confused a little bit on example 9 because I didn’t understand how they got y-0=(-1)(x-2) but then I realized they subtracted x & y from the coordinates (2,0). Besides that the textbook was good at describing the examples they showed. The graphs helped alot because being able to see it visually made me remember the material.

Right, Example 9 is using the “point-slope form” of the equation of a line:

y-y_1 = m(x-x_1)

If you know the slope of the line and a point on the line, the equation can be determined from above by replacing m with the slope, and replacing x_1 and y_1 with the coordinates of the point.

1.2 was a better section. It was a great review to see the kinds of trends in graphs for exponential, algebraic and logs and rational functions. The figure 18 about trigonometric functions was a little confusing and I look forward to learning more about sin and cos to clear up the confusion in class. I especially thought the information about the 1 degree, 2 degree an third and fourth degree polynomials was helpful.

The textbook did well defining a function in a simple way that allows one to understand exactly what it is along with providing helpful examples. The graphs helped me to better visualize what the paragraphs were explaining especially on page 16 where it talks about the line x=-1 coinciding with the line y=-1.It helped elaborate more on what was discussed in class.

I thought that it was interesting to think of a function as a machine. With X being the “input” and F(x) being the “output”. That really helped me understand the concept better. It was nice to review the section and get a better understanding of the definitions. The graphs and examples really helped as well.

The first part of the reading assignment did well to compliment todays classwork. The concepts presented in the piecewise section were more difficult to grasp but made sense when combined with the graphs and especially with the “realworld” situation in example 10. I can see a grading scale also being represented as a step function.

Yes, exactly. In this course, for example, at a grade of 93% you would see a jump in letter grade from A- to A.

Pages 10-12 were basically refresher pages for me. I had actually heard of thinking of a function as a machine because of my high school Pre-Calc teacher. I think that is a very good way to think about functions and how they work. I understood all four of the situations presented for the ways to represent a function. The layout of the textbook and the examples seem to be understandable for me with this reading.”Piecewise defined function” didn’t ring a bell for me from high school, but looking at the book’s explanation and examples on pages 16-17 I understand it. It was helpful that they put the point-slope form on the side of page 17. Otherwise I would have had to have looked it up to fully understand the example.

I already took MAT 109 so I pretty much already understand the idea of functions and how they work but the section 1.1 on the Piecewise Defined Functions was something i struggled with in 109 because i didn’t get a full understanding of the concept due to not being able to see what portion of what goes with what function.But when it comes to the graphs on the function I have a pretty basic understanding of it.

Hopefully the tax bracket example helped with your understanding of piecewise functions. It is basically a function that is made up of “pieces” of other functions. In the tax bracket example, the function T(x) is made up of pieces of three lines.

I haven’t taken a math course since my junior year of high school and I never thought I could forget so much in just four years. The discussion in class today and reading the text book definitely helped refresh my mind and make that time period seem a little shorter (maybe 3.5 years lol.) I especially liked how the book offers so many examples and writes out all the steps so it’s easy to follow along. The analogy of a function being similar to a machine helped clarify the idea of inputs and outputs and how they can be related in these equations. Domain/range and constructing a box were surely a confidence booster being some of the rare information I do still remember. I found myself making several little mistakes like how to add or multiply exponents but I’m sure that as the semester progresses I’ll become more confident in how to handle these situations.

Any time you have one variable depend on another. Some expression where each input â€śdomainâ€ť (x) has on unique output â€śrangeâ€ť f(x) . Domain are all values of x for which the function is defined and range are all the values of f(x). f the domain is called an independent variable and the range is called a dependent variable. See in it as a machine that basically takes inputs to come out with specific outputs is a good way of viewing how function works. The graph is the most practical way to view the relations of a domain to a range to see the whole picture. The independent variable are the inputs and the dependent variables are the outputs .On page 12 it discusses what difference quotient of a function , it has it uses for average range of change and slopes of tangent lines. Denoted as f(a + h) â€“ f(a)/h. Representations of Functions by four possible ways such as verbally, numerically, visually and algebraically .On page 16 piecewise defined functions that is defined in several equations. The use of absolute value is graphed in a V shape. Whatever is in the absolute value bracket you do the opposite of the either positive or negative to the slope get a vortex. Step functions is seen as horizontal line on a graph. This is a major refresher for me since my last math class was taken several years back.

I really enjoyed this section about functions. In high school when I took Pre-Calculus class this was the section that I mostly understood. But the only part that I never fully understood is how to use the peace-wise functions, like knowing which one to use and then knowing how to get a domain and range when graphing.

Hopefully the tax bracket example helped a little bit with your understanding of piecewise-functions. In terms of using them, each piece of a piecewise function has a range of x-values that it’s used for. For example, in the tax bracket example, we use the first part .15x only when 0<= x <= 19,300.

In terms of graphing piecewise functions, one way to think about doing it is this: draw the graphs of all of the pieces, and then "erase" the parts that you don't need. For example, for the tax bracket function, you could start drawing the graph of the line y=.15x. Since this "piece" is only used when 0<= x <= 19,300, you would then erase the part of the line to the left of x=0 and the part of the line to the right of x=19,300. So you're are just leaving behind a piece of the line. Similar idea for the other two pieces.

Anthony Rebmann

This was mostly a summary of what you talked about in class today. Except for the few terms that were not mentioned by you such as absolute value, piecewise defined function, even function, etc. They talked about how there are four ways to express a function. They went into detail about graphing them and had examples. They thoroughly explained what a function is and how it works.