But they do give us the denominator and so, we can think about what are the interesting numbers, what are the interesting x-valuesįor the denominator. Try to engage in the problem as opposed to just watch me do it. I start to think about it with you, pause it anytime, It out or if you were having trouble with it as And like always, pause the video, and see if you can figure The fourth choice is off right over here. Which of the following is a possible graph of y equals f of x? And they give us four choices. Squared minus x minus six, where g of x is a polynomial. We're told, let f of x equal g of x over x Instead we should look for either two asymptotes at the correct spots or two empty dots at the correct spots or one of each. So, to answer your final question, in this specific example, we cannot tell which would happen without seeing the numerator. If the binomial factor remains in the denominator because it cannot be cancelled, it will show up as a vertical asymptote on the graph at the value of x that would be undefined. If so, it would be an empty circle on the graph (shows up as ERROR on table list of calculator), which means it is a removable discontinuity. IF we could see the numerator, we could find out whether each of the binomial factors cancels out with a corresponding binomial factor in the numerator. So, what happens when the denominator of a rational function becomes zero? It either just has a gap at that point (indicated by an empty dot, or little circle), or it has a vertical asymptote at that point which means that the function swoops toward positive or negative infinity, never crossing that point. If the denominator becomes zero then the function is undefined at THAT point - that value of x. What Sal is saying is that the factored denominator (x-3)(x+2) tells us that either one of these would force the denominator to become zero - if x = +3 or x = -2. The denominator of a rational function can't tell you about the horizontal asymptote, but it CAN tell you about possible vertical asymptotes. We have no information about the numerator. In this case, the only detail we have is that there is a quadratic in the denominator. With a little practice, though, you can figure out a lot about a graph by looking at the parts of these rational functions. It isn't like the equation of a line, (linear function), f(x) = mx +b, where you just have slope and intercept to worry about. At first, rational functions seem wildly complicated. Government Agencies: Government agencies need to understand the financial viability of projects and programs and they use break-even analysis to determine this.This example is a question about interpreting the parts of expressions.Businesses: A broad range of businesses use break-even analysis to paint a better picture of their cost structure, pricing, as well as their operational efficiencies.This analysis help them determine how much money to allocate a transaction and which assets would generate the higher profits for them. Stock and Option Traders: Break-even analysis is crucial for stock and option traders because they need to know how much money is needed to cover their expenses for each transaction they make.With this information they make more informed decisions on their asset selections. Investors: Investors conduct break-even analysis to determine the financial performance of companies.Financial Analysts tie break-even analysis into their valuations and recommendations on a business. Financial Analysts: These professionals use break-even analysis as a profitability and risk metric.This is critical for the early stage of a business. Entrepreneurs: Break-Even analysis is useful for entrepreneurs and founders because it helps determine the minimum level of sales needed to cover costs.
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