1/04/2023

What happens if we graph both f and f^{-1} on the same set of axes, using the x-axis for the input to both f and f^{-1} ?

 What happens if we graph both f and f^{-1} on the same set of axes, using the x-axis for the input to both f and f^{-1} ? 

Suppose f:R \rightarrow R is a function from the set of real numbers to the same set with f(x)=x+1. We write f^{2} to represent f \circ f and f^{n+1}=f^n \circ f. Is it true that f^2 \circ f = f \circ f^2? Why? Is the set {g:R \rightarrow R l g \circ f=f \circ g} infinite? Why? 

Inherently, the range of a function f (x) is the domain of the inverse function f^{-1}. The domain of f  is the range of  (f^{-1} \circ f)(x)=f^{-1}(f(x))=f^{-1}(y)=x  So, what happens if we graph both f and f^{-1} on the same set of axes, using the x-axis for the input to both f and f^{-1} ? Let's try some examples.

If we type y=x^3  \lbrace{-2 < x < 2}\rbrace y=x^{1/3}  \lbrace{-2 < x < 2}\rbrace , and  y = x   \lbrace{-2 < x < 2}\rbrace  on the graph, all of the three functions will be limited in the same input   \lbrace{-2 < x < 2}\rbrace    and their output will also affected by such a input boundary. The domain of f  become the domain of  (f^{-1} \circ f)(x)  , not its range. 




Let's try another example: C° to F°: Celsius to Fahrenheit Conversion Formula

 F = C×( \frac{9}{5} )+32

 F^{-1} = (\frac{9}{5})(C-32)





Again, as we can see that these these three lines are more like projections of each other from different angles rather than reflection.

The difficulty of setting the same set of axes for f and f^{-1} is that f^{-1} will become not fully or true inverse function of f  since the output of f^{-1} is limited. Instead of setting the input {-100 < x < 100}, the input of  f^{-1}  should be the range of f 


Suppose f:R \rightarrow R is a function from the set of real numbers to the same set with f(x)=x+1. We write f^{2} to represent f \circ f and f^{n+1}=f^n \circ f

When f^{n+1}=f^n \circ f.  and  n \geq1

 f^2=f \circ f

If f^{2} represents f \circ f  =f(f(x))  ,  f^2 \circ f =f^2(f(x))  
 f^2=f(x+1)=(x+1)+1=x+2

It is similar to an arithmetic progression which the next one is the previous number +1

Recall that  f^2=f \circ f=f(f(x))   
 f^2 \circ f=f(f(f(x)))
 f \circ f^2=f(f^2(x))=f(f \circ f)=f(f(f(x)))
Therefore,  f^2 \circ f = \( f \circ f^2 = \( f(f(f(x)))

 g \circ f =g(f(x)) = f \circ g=f(g(x))
In this case,  f(x)=x+1    g(f(x))=g(x+1)=g(x)+1=f(g(x))
Now, we know that the set has to satisfy the equation  g(x+1)=g(x)+1  
If the domain of  g(x)  is all real numbers, then g(1)=g(0)+1, g(2)=g(1)+1, g(3)=g(2)+1,....... it will be infinite.

Polynomial and Rational Functions

 A gold bar maker wants to design a 14-kg gold bar for its customers. Assume the density of gold is 1 kg/m³, the height of the bar is x, the length is designed to be 3x+4, and the width is x+1, how do we know the height of the bar should be?

The weight of the bar equals the volume of the bar times the density of gold. The volume of the bar equals is the product of its height, length, and its width. Therefore, we have....

  x(3x+4)(x+1)1=14

  x(3x^2 +7x +4)=14

 3x^3 +7x^2 +4x -14=0

The factor of  3x^3  is   \pm1 \pm3

The factor of  -14  is   \pm1 \pm2 \pm7

Test these factors start with 1...

 3(1)^3 +7(1)^2 +4(1) -14=0  ,  fortunately, we have a real zero at x=1

 3x^3 +7x^2 +4x -14=(x-1)(3x^2 +10x +14)=0



On the graph, there is only one x-intercept at x=1
However, the quotient  3x^2 +10x +14  does not not have any intercept at y=0


10/23/2022

Which Ethical and Social Responsibilities Do Companies Have When It Comes To Quality?

 Corporate Social Responsibility (CSR)

Social responsibility requires businesses must act in a manner benefiting society, rather than just the bottom line. It has become increasingly important to investors and consumers who seek investments that not only are profitable but also contribute to the welfare of society and the environment. Corporate social responsibility (CSR) is a self-regulating business model that helps a company be socially accountable to itself, its stakeholders, and even the planet. 


Business Ethics

Companies usually adopt principles of corporate ethics that are imposed by legislation or by itself such as codes of conduct that apply to their employees. Business ethics is what drives that code of conduct. And it also shapes the way companies are seen by the world. Basically, it makes sure a company operates according to all applicable laws and maintains the company's respect among its peers and potential clients or customers.


Which Ethical and Social Responsibilities Do Companies Have When It Comes To Quality?

There are numerous effective quality tools have been used by industry for decades to reduce waste and improve efficiency, but they have not been widely recognized in the corporate social responsibility (CSR) space. Quality is generally based on a set of values and beliefs at its center such as zero waste and maximizing customer satisfaction. Similar to CSR, quality management also has a very strong focus on people, not just customer satisfaction, but also the quality of working life such as employee satisfaction. 


In addition to a strong link in the core values and concepts, CSR and quality management also share an interest in several common issues such as waste reduction, safety, supplier accountability, and empowerment. For CSR, waste minimization and pollution prevention address key issues related to resource use, energy, and significant environmental trends that affect a wide range of stakeholders. 


There is a strong correlation between quality and ethics as well. When leadership demonstrates its philosophy and practice of ethical behavior, it impacts the whole organization, whether in education, government, or commercial enterprises. ISO9000 and the EFQM Model both emphasize heavily on leadership. Great leaders put ethics, quality, effectiveness, and respect for people to eliminate fear in the system, and they look in the mirror often to see if their own thinking is ethical and a cause of quality throughout the system. 


The Consequences of Poor Quality

Poor quality can lead to considerable negative outcomes that are associated with customers, daily operations, or society. Obviously, poor quality can cause customers to take their business elsewhere, leading to job losses and a decrease in company productivity. For example, back in 2011, HTC was the third biggest phone manufacturer by global market share, only behind Samsung and Apple. Poor quality also has an impact on employee productivity. Employees working on products or services of poor quality may be less productive and may experience stress due to the lack of quality assurance. HTC's smartphones were the go-to Android devices and offered an extremely good price-to-performance ratio. However, it took hundreds of millions of years for dinosaurs to evolve, but it takes only a few years for a hot smartphone brand to turn into a dinosaur. That’s what happened to HTC.


Moreover, poor quality can have several adverse effects on society as a whole. These can include increased costs, decreased efficiency, and even harm to people and the environment. For example, the reason pharmaceutical drugs are inherently safer than nutritional supplements is probably due to the fact drugs produced by the pharmaceutical industry are regulated by the FDA. If all these drugs are inadequately produced, how many people will be affected? During the pandemic of COVID-19, it was difficult to ensure the supply of necessary hygiene products, including disinfectants. In many countries, therefore, the framework conditions for production and marketing were eased. The FDA's concerns for quality and safety do not only include hand disinfectants. Since the beginning of the pandemic, the disinfection of surfaces has also played a more important role than before. However, not all devices on the market offer the required safety standard for the user.


Quality, Responsibility, or Maximize Profits for Shareholders?

I reckon that all businesses have certain ethical and social responsibilities that they have to fulfill because of operating in society and using society's resources to produce their products and services. Ethical responsibility includes showing moral characteristics such as providing labor insurance and caring for the environment. Businesses have a social and ethical responsibility to maintain the quality of their products as poor quality can harm the health of the people and the environment. For example, selling expired or unhygienic food can be lethal for humans consumptions and customers can be both the shareholders and stakeholders of the company. Businesses' demand for superior quality can be considered a response to both categories of profit-making and social responsibility. In Taiwan, Ting Hsin Oil Company made illegal profits of more than 100 million NTD by importing low-quality oil products and processing them into edible oil from 2012 to 2014. Chairman Wei Yingchong was charged with violating the Food Safety and Hygiene Management Law and sentenced to four years in prison by the judge.



Reference

Consequences of poor quality. Quality Gurus. (2022, September 5). Retrieved October 23, 2022, from https://www.qualitygurus.com/consequences-of-poor-quality/ 


ECA Academy. (2022, August 30). FDA draws consequences for poor quality of hand disinfectants and UV wands. ECA Academy. Retrieved October 23, 2022, from https://www.gmp-compliance.org/gmp-news/fda-draws-consequences-for-poor-quality-of-hand-disinfectants-and-uv-wands 


Ganti, A. (2022, September 8). Social Responsibility in business: Meaning, types, examples, and criticism. Investopedia. Retrieved October 22, 2022, from https://www.investopedia.com/terms/s/socialresponsibility.asp 


Indeed Editorial Team. (n.d.). The ethics of Quality. The W. Edwards Deming Institute. Retrieved October 23, 2022, from https://deming.org/the-ethics-of-quality/ 


Sapru, R., & Schuchard, R. (n.d.). CSR and quality - home | BSR. CSR and Quality: A Powerful and Untapped Connection. Retrieved October 23, 2022, from https://www.bsr.org/reports/BSR_ASQ_CSR_and_Quality.final.pdf 

10/18/2022

2022 registration trends for ISO9000

 ISO 9001 sets out the criteria for a quality management system and It can be used by many large and small organizations. The standard is based on several key quality management principles such as customer focus, leadership, and process approach. Businesses are using ISO 9001 to ensure that their customers get consistent, good quality products and services, which in turn brings many business benefits. Nowadays, it has adapted to various sectors like manufacturing, technology, and even the healthcare industry.


In the past few years, the global economic situation and the practice of the business community pushed ISO to expedite the finalization of the international standard for risk management. Due to the pandemic of COVID-19, the certification is more focused on the area of risk management. Because of this reason, more and more companies are getting their ISO 9000 certification done, meaning that the registration trend is positive. In addition, Organizations are showing interest now more than ever, in getting their ISO 9000 certification done since it is one of the most effective tools of quality management available in the market. Since workplace safety has become more and more emphasized, high-risk industries such as construction and basic metal production have shown the highest level of registration. With the need for more stringent safety requirements, ISO 9000 certification provides the all-in-one solution to follow. Moreover, not only western countries but also countries in the East have increased the rate of registration for ISO 9000 certification. 


In conclusion, I believe that risk can further be described as a situation involving exposure to danger and also refers to the result of uncertainty in situations that can be avoided through pre-emptive action. If organizations are able to ensure the risk doesn’t threaten the continuity of the organization, they are more likely to thrive in the long run. There are two most recent ISO9000 trends, risk management and eastern countries are increasing their registrations.




Reference

ISO 9000 family - quality management. ISO. (2020, March 11). Retrieved October 17, 2022, from https://www.iso.org/iso-9001-quality-management.html 


Risk management and based thinking in ISO 9001. ISO Global. (n.d.). Retrieved October 17, 2022, from https://isoglobal.com.au/articles/risk-management-iso-9001/ 

10/10/2022

How can your organization benefit from organizational learning? How would you implement it?

I think that before we discuss how we benefit from organizational learning, we should understand why we are doing organizational learning first. Briefly, organizational learning allows a business to continue to advance its professional growth and align its employees as a team. As the company improves, it develops and accumulates a broad range of knowledge and information for its future. Now, we know the advantages. Let's look at how to implement it. 


There are many vital aspects we must consider such as leadership, customers, alignment, widespread, support, measurement, and communication. Based on this concept, I believe that I have to plan an organizational learning program. Therefore, I sorted these factors into interconnected principles. Starting with customer satisfaction and leadership, I believe that developing and clearly communicating the desired vision is the first step. Then, develop the measurement system, and align and support the learning processes. After that, widespread the story and keep it in the policies and management. For example, Dyson, a name that has become synonymous with vacuum cleaners and is known for transforming a brilliant engineering idea into a bright success. The consistency in the open culture is embedded in the very fabric of the organization where mistakes are not punished but rather valued because of the potential to be turned into success. In addition, 3M’s unique 15% Culture encourages employees to set aside a portion of their work time to proactively cultivate and pursue innovative ideas that excite them. Whether it’s experimenting with new technology, forming a special interest group around a fresh idea, or finding a new way to run a process, its 15% Culture gives employees in all areas the license to innovate. 


Two years ago, when I worked at a coffee shop, I found that the barista is more likely to actively practice and make better latte art when they feel valued and respected. Employees can do one thing as good as 60% or as good as they can to make it close to 99%. I think it is the core of effective learning, so I discussed this observation with my boss and provided my recommendations. She(the boss) accepted my idea and began to share their art stories. Moreover, I suggested a Tasty Program. In this program, everyone can discover and find their favorite formula of drink with the support of the shop. And, they can directly benefit from the revenue generated by the formula which means a proportion of the profit per sup of the drink sold. Each week, the most popular top three will be listed in a specific area of the menu called, The Barista, "the innovator's name" series. 




Reference

How leaders can change company culture: Lessons from 3M and Six sigma. Network for Business Sustainability (NBS). (2022, February 21). Retrieved May 14, 2022, from https://nbs.net/other/the-challenge-of-change-3m-six-sigma-and-corporate-culture/


Indeed Editorial Team. (2021, July 7). Career development | indeed.com. What Is Organizational Learning and Why Is It Important? Retrieved October 9, 2022, from https://www.indeed.com/career-advice/career-development


Ltd, A. A. (2021, December 31). Dyson Company Analysis: Industry and Culture. UK Essays. Retrieved February 1, 2022, from https://www.ukessays.com/essays/business/examining-the-rise-of-the-dyson-company-business-essay.php


Quality Management (Knowles) Ch 13 p 150-166, Ch 15 p 177-185

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