answer:When solving equations, it's common to divide both sides by a variable expression to simplify or isolate a particular term. However, this operation can sometimes lead to pitfalls, especially if the variable expression can take on zero values. When you divide both sides of an equation by a variable expression, you're implicitly assuming that the expression is non-zero. If the expression can be zero, you're essentially dividing by zero, which is undefined in standard arithmetic. This can lead to the following issues: 1. **Loss of solutions**: If the variable expression can be zero, dividing both sides by it may eliminate valid solutions from the original equation. These solutions are often referred to as extraneous solutions or singular solutions. By dividing by the variable expression, you're essentially excluding these solutions from the resulting equation. 2. **Introduction of extraneous solutions**: Conversely, dividing both sides by a variable expression can also introduce new solutions that don't satisfy the original equation. These extraneous solutions may arise from the fact that the division operation is not valid when the expression is zero. 3. **Reducing the solution set**: Dividing both sides by a variable expression can reduce the solution set of the original equation. This is because the resulting equation may not be equivalent to the original equation, especially if the variable expression can be zero. To illustrate these pitfalls, consider the simple equation x^2 = x. If we divide both sides by x, we get x = 1, which seems to imply that the only solution is x = 1. However, this division is only valid when x ≠ 0. The original equation x^2 = x also has the solution x = 0, which is lost when we divide both sides by x. In the context of the original problem we solved, dividing both sides by f(x) or f(y) did indeed assume that these expressions are non-zero. However, as we discussed, the condition f(1) = 1 ensured that f(x) cannot be identically zero. Nevertheless, it's essential to be aware of the potential pitfalls of dividing both sides of an equation by a variable expression and to carefully consider the implications of such operations on the solution set. To avoid these issues, it's often helpful to: * Check for potential divisions by zero and consider the implications on the solution set. * Verify that the resulting equation is equivalent to the original equation. * Be cautious when dividing both sides of an equation by a variable expression, especially if the expression can be zero. * Consider alternative methods, such as factoring or using algebraic manipulations that avoid divisions by variable expressions.

answer:Let's analyze the functional equation f(x+y) = f(x)*f(y) and see how it leads to the form f(x) = e^(c*x). First, we define a new function g(x) = f(x) when x is an integer, specifically g(n) = f(n) for all integers n. We also define a constant c such that f(1) = e^c. We don't know the value of c yet, but we'll see how it relates to f(x) later. Since f(x+y) = f(x)*f(y) holds for all real x and y, we can choose x = y = 1 to get f(2) = f(1)*f(1) = e^(2c). Similarly, we can show that f(3) = f(2)*f(1) = e^(3c), and in general, f(n) = e^(nc) for all positive integers n. We can extend this result to all integers n by using the relation f(x+y) = f(x)*f(y) with x = n and y = -n. This gives us f(0) = f(n)*f(-n), which implies that f(-n) = 1/f(n) = e^(-nc) for all positive integers n. So far, we have determined that f(n) = e^(nc) for all integers n. To extend this result to all rational numbers, we can use the relation f(x+y) = f(x)*f(y) with x = y = 1/n, where n is a positive integer. This gives us f(2/n) = f(1/n)*f(1/n) = e^(2c/n), and in general, f(m/n) = e^(mc/n) for all integers m and n. Now, we need to extend this result to all real numbers x. To do this, we can use the fact that f(x) is continuous for all real x. Specifically, we can choose a sequence of rational numbers r_n that converges to x, and use the continuity of f(x) to show that f(x) = lim f(r_n) = lim e^(cr_n) = e^(cx). The reason for this is that the exponential function e^(cx) is continuous for all real x and c, and the limit of a sequence of exponential functions is also an exponential function. Therefore, the only continuous function that satisfies f(x+y) = f(x)*f(y) for all real x and y is of the form f(x) = e^(cx), where c is a constant. Finally, the condition f(1) = 1 implies that e^c = 1, which means that c = 0. Therefore, the only continuous function that satisfies f(x+y) = f(x)*f(y) for all real x and y and also satisfies f(1) = 1 is f(x) = e^(0x) = 1. In summary, the functional equation f(x+y) = f(x)*f(y) leads to f(x) being of the form f(x) = e^(cx) because of the way that the exponential function arises from the repeated application of the functional equation. The continuity of f(x) and the condition f(1) = 1 then determine the value of c, which in this case is c = 0.

answer:The Bayonet Constitution of 1887 was a significant document in Hawaiian history that dramatically altered the country's government and power structure. Its primary purpose was to limit the authority of King Kalākaua and transfer power from the monarch to the legislature and the cabinet, which was dominated by American and European businessmen. This constitution was forcibly imposed on the King, earning its name due to the armed militia present during its signing. The document stripped the monarch of much of his executive power, giving more control to the foreign-dominated legislature, and effectively disenfranchising many Native Hawaiians and immigrant laborers by introducing property qualifications for voting. Ultimately, the Bayonet Constitution contributed to the overthrow of the Kingdom of Hawaii in 1893, when American and European business interests seized control of the islands, leading to the eventual annexation of Hawaii by the United States in 1898. The constitution remains a contentious and painful chapter in Hawaiian history, symbolizing the loss of sovereignty and the erosion of native Hawaiian rights and self-governance.

answer:The Bayonet Constitution was forced on King Kalākaua in 1887 by a group of American and European businessmen, who were primarily sugar planters, to consolidate their power and control over the islands. They sought to reduce the monarch's authority and create a government more aligned with their own economic interests. One of the main reasons behind this move was the king's attempts to limit the power of the foreign businessmen who dominated the sugar industry in Hawaii. King Kalākaua had been trying to assert his authority and reduce the influence of these outsiders, which made them feel threatened. They were concerned about the king's efforts to regain control over the land, which they had acquired through questionable means, and his attempts to limit their economic power. Another factor was the king's support for the native Hawaiian people and his efforts to preserve their rights and way of life. The foreign businessmen saw the native Hawaiians as a workforce to be exploited, and they opposed any efforts to improve their conditions or give them more control over their land. By forcing the Bayonet Constitution on the king, the businessmen aimed to disenfranchise many native Hawaiians and immigrant laborers, thus maintaining their power and control over the islands. The final trigger for the overthrow was the king's proposal to lease Pearl Harbor to the British, which would have undermined American influence in Hawaii. This move, combined with the king's attempts to assert his authority, led the American and European businessmen to take drastic action and impose the Bayonet Constitution, marking a significant turning point in the history of Hawaii.