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Determine and simplify the indicated composite function and state its domain using interval notation. Create a sign diagram for the function \(f(x) = x^3 + 5x^2 - 12x - 36\), and use a calculator to help sketch its graph. Evaluate the logarithmic expression \(\log_{10} 107 3\). Find the inverse of the function \(f(x) = 5x / (x - 4)\), check your answer algebraically and graphically. Using the half-life of Carbon-14 (5730 years), calculate the percentage of the original amount remaining after 30,000 years, rounding to the nearest tenth.

In addition, reflect on writing a body paragraph for an academic paper. Ensure the paragraph includes a topic sentence supporting the thesis statement, one direct quote, and one paraphrased passage with proper APA in-text citations. Explain how these elements support your claims and add credibility. Include the original source text for the paraphrase and an APA-formatted reference list for all sources used.

Paper For Above instruction

The tasks presented involve a variety of mathematical and academic writing skills, starting with the simplification of composite functions and analysis of their domains. For example, consider the composite function \((h \circ f)(x) = h(f(x))\), where understanding the individual functions and their domains is critical. When simplifying such functions, it is essential to evaluate the composition carefully, ensuring that the domain of the resulting function accounts for restrictions from both functions involved. For the given functions \(f(x) = 3x - 6\), \(g(x) = |x|\), \(h(x)\), and \(k(x) = x / (x - 1)\), the composite \(h(f(x))\) would involve substituting \(f(x)\) into \(h\) and then considering the domain restrictions accordingly. Calculating this and expressing the domain in interval notation enhances understanding of the behavior of these functions.

Next, creating a sign diagram for the cubic function \(f(x) = x^3 + 5x^2 - 12x - 36\) involves first finding critical points where the function’s derivative equals zero, then analyzing the sign of the function in the intervals determined by those points. Using a calculator to sketch the graph helps visualize the function's increasing and decreasing behavior, as well as the locations of roots and turning points. Sign diagrams and graph sketches deepen comprehension of polynomial functions and their characteristics, which are essential concepts in calculus and algebra.

Evaluating logarithmic expressions such as \(\log_{10} 107\) is another fundamental skill. Applying properties of logarithms and calculator functions allows for precise computation, which is critical in fields such as engineering and sciences where logarithmic scales are common. Accurate evaluation of logarithms supports problem-solving involving exponential growth or decay, thus connecting algebraic skills with real-world applications.

Finding the inverse function of \(f(x) = \frac{5x}{x - 4}\) involves algebraic manipulation: swapping \(x\) and \(y\), solving for \(y\), and then verifying the result both algebraically and graphically. Specifically, solving for \(y\) reveals the inverse \(f^{-1}(x) = \frac{4x}{5 - x}\). Verification steps confirm the inverse's correctness, demonstrating understanding of inverse functions’ properties and their graphical counterparts.

In real-world scenarios, radioactive decay calculations utilize exponential models based on half-lives. The half-life of Carbon-14 is approximately 5730 years. Using the decay formula \(N(t) = N_0 e^{-\lambda t}\), with \(\lambda = \frac{\ln 2}{\text{half-life}}\), the percentage remaining after 30,000 years can be calculated, illustrating the application of mathematics to scientific problems. Rounding to the nearest tenth provides a clear and accurate estimate, vital for scientific reporting.

Complementing mathematical analysis, academic writing skills are critical for effective communication. Developing a cohesive body paragraph that supports a thesis involves crafting a clear topic sentence, integrating direct quotes and paraphrased material with proper APA citations. Explaining how these elements bolster the thesis and lend credibility demonstrates critical thinking and adherence to academic integrity standards. Including the original source text and creating a comprehensive reference list ensures transparency and scholarly rigor.

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