Solving Composite Functions And Trigonometric Equations A Step By Step Guide

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In the realm of mathematics, composite functions and trigonometric equations form integral components of mathematical analysis. Composite functions involve the application of one function to the result of another, creating a chain of operations. Trigonometric equations, on the other hand, explore the relationships between angles and their trigonometric ratios. This article delves into a specific problem that combines these concepts, requiring us to find the value of 'x' under different conditions involving composite functions and trigonometric equations. We will explore how to solve composite functions, evaluate trigonometric expressions, and ultimately, find the solutions that satisfy the given conditions. Let's embark on this mathematical journey, unraveling the intricacies of functions and trigonometry.

To begin, let's define the functions provided in the problem. We have g(x) = 2x + 2 and (f ∘ g)(x) = 8x + 13. The notation (f ∘ g)(x) represents the composite function, which means f(g(x)). In simpler terms, we first apply the function 'g' to 'x' and then apply the function 'f' to the result. Our goal is to find the expression for f(x). To achieve this, we will use the given information about (f ∘ g)(x). Since (f ∘ g)(x) = f(g(x)), we can substitute the expression for g(x) into this equation. This gives us f(2x + 2) = 8x + 13. Now, to find f(x), we need to make a substitution that will isolate 'x' inside the function 'f'. Let's substitute y = 2x + 2. Solving for 'x', we get x = (y - 2) / 2. Now we can rewrite the equation f(2x + 2) = 8x + 13 in terms of 'y'. Substituting x = (y - 2) / 2 into the equation, we get f(y) = 8((y - 2) / 2) + 13. Simplifying this expression, we have f(y) = 4(y - 2) + 13, which further simplifies to f(y) = 4y - 8 + 13, and finally, f(y) = 4y + 5. Replacing 'y' with 'x', we obtain the expression for f(x) as f(x) = 4x + 5. This is a crucial step in solving the problem as it allows us to work with the individual functions and their properties. With f(x) now defined, we can move on to the next part of the problem, which involves trigonometric functions and composite functions in reverse order.

Next, we are tasked with finding the value of 'x' when cot(x) = 20. The cotangent function, denoted as cot(x), is the reciprocal of the tangent function, tan(x). Therefore, cot(x) = 1 / tan(x). The tangent function, in turn, is defined as the ratio of the sine function to the cosine function, tan(x) = sin(x) / cos(x). Thus, cot(x) = cos(x) / sin(x). Given that cot(x) = 20, we have cos(x) / sin(x) = 20. To find the value of 'x', we first need to find the value of tan(x). Since cot(x) is the reciprocal of tan(x), we have tan(x) = 1 / cot(x) = 1 / 20. Now, we can use the inverse tangent function, also known as arctangent and denoted as arctan(x) or tan⁻¹(x), to find the principal value of 'x'. The arctangent function gives us the angle whose tangent is a given number. So, x = arctan(1 / 20). Using a calculator, we find that x ≈ 0.0499 radians. However, the tangent function has a period of π, which means that tan(x) = tan(x + nπ) for any integer 'n'. Therefore, there are infinitely many solutions for 'x'. The general solution for 'x' is given by x = arctan(1 / 20) + nπ, where 'n' is an integer. This means that the values of 'x' that satisfy cot(x) = 20 are spaced π radians apart. For practical purposes, we often consider the principal value, which lies in the interval (-π/2, π/2). In this case, the principal value is approximately 0.0499 radians. Now that we have found the value of 'x' for the given trigonometric equation, we can move on to the final part of the problem, which involves evaluating the composite function in the reverse order.

The final part of the problem asks us to find the value of 'x' when g(f(x)) = 20. This is the composite function in the reverse order compared to what we were given initially. We already found the expression for f(x), which is f(x) = 4x + 5. Now we need to find g(f(x)). We know that g(x) = 2x + 2, so to find g(f(x)), we substitute f(x) into the expression for g(x). This gives us g(f(x)) = 2(f(x)) + 2. Now we substitute the expression for f(x): g(f(x)) = 2(4x + 5) + 2. Simplifying this expression, we get g(f(x)) = 8x + 10 + 2, which further simplifies to g(f(x)) = 8x + 12. We are given that g(f(x)) = 20, so we can set up the equation 8x + 12 = 20. To solve for 'x', we first subtract 12 from both sides of the equation: 8x = 20 - 12, which gives us 8x = 8. Then, we divide both sides by 8: x = 8 / 8, which gives us x = 1. Therefore, the value of 'x' that satisfies g(f(x)) = 20 is x = 1. This completes the solution to the problem. We have successfully found the value of 'x' under both conditions: when cot(x) = 20 and when g(f(x)) = 20. The first condition required us to use trigonometric identities and the inverse tangent function, while the second condition involved evaluating a composite function in reverse order. These are fundamental concepts in mathematics, and understanding them is crucial for solving more complex problems.

In conclusion, this problem has provided a comprehensive exercise in understanding and applying concepts related to composite functions and trigonometric equations. We started by unraveling the composite function (f ∘ g)(x) to find the expression for f(x). This involved substituting and simplifying expressions, a fundamental skill in algebra. Then, we tackled the trigonometric equation cot(x) = 20, which required us to use the reciprocal relationship between cotangent and tangent, as well as the inverse tangent function. We also discussed the periodic nature of the tangent function and the general solution for 'x'. Finally, we evaluated the reverse composite function g(f(x)) = 20, which involved substituting f(x) into g(x) and solving the resulting linear equation. The solution to this problem demonstrates the interconnectedness of different areas of mathematics. It highlights how algebraic manipulation, trigonometric identities, and the properties of functions can be combined to solve complex problems. Mastering these concepts is essential for further studies in mathematics and related fields. The ability to work with composite functions and trigonometric equations is not only valuable in academic settings but also in various real-world applications, such as physics, engineering, and computer science. By understanding the underlying principles and practicing problem-solving techniques, one can develop a strong foundation in mathematics and confidently tackle a wide range of challenges.

  • Composite functions
  • Trigonometric equations
  • Cotangent function
  • Inverse tangent function
  • Function composition
  • Solving equations
  • Mathematical analysis
  • Trigonometry
  • Algebra
  • Functions