Finding F⁻¹(4) Given Functional Equations - A Step-by-Step Guide

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This comprehensive guide delves into the process of finding the inverse function value, specifically f⁻¹(4), for two distinct function scenarios. We will explore the methodologies involved in determining the inverse function and subsequently evaluating it at a specific point. Understanding inverse functions is crucial in various mathematical contexts, including solving equations, analyzing function behavior, and applications in calculus and other advanced topics. This exploration provides a step-by-step approach to solving these types of problems, ensuring a clear understanding of the underlying concepts.

(b) If f(x + 5) = 10x + 11, find f⁻¹(4)

In this section, we tackle the problem of finding f⁻¹(4) given the function f(x + 5) = 10x + 11. The core idea here is to manipulate the given functional equation to express x in terms of f(x + 5), which will then lead us to finding the inverse function. This process involves a series of algebraic steps and a clear understanding of function composition and inverse functions. The following detailed explanation breaks down each step, ensuring clarity and comprehension.

First, let's introduce a substitution to simplify the given function. Let y = x + 5. This substitution allows us to rewrite the function in a more manageable form. Substituting y into the equation f(x + 5) = 10x + 11, we get f(y) = 10x + 11. However, we need to express x in terms of y to have f(y) solely in terms of y. From the substitution y = x + 5, we can isolate x by subtracting 5 from both sides, yielding x = y - 5. Now, substitute this expression for x back into the equation f(y) = 10x + 11. This gives us f(y) = 10(y - 5) + 11. Expanding this, we get f(y) = 10y - 50 + 11, which simplifies to f(y) = 10y - 39.

Now that we have f(y) expressed in terms of y, we can find the inverse function. To find the inverse function, we typically swap y and x and solve for y. So, let's rewrite f(y) = 10y - 39 as x = 10y - 39. Next, we solve for y. Add 39 to both sides to get x + 39 = 10y. Then, divide both sides by 10 to isolate y, resulting in y = (x + 39) / 10. This expression for y represents the inverse function, which we denote as f⁻¹(x) = (x + 39) / 10. We have now successfully found the inverse function.

Finally, to find f⁻¹(4), we substitute x = 4 into the inverse function we just derived. So, f⁻¹(4) = (4 + 39) / 10. This simplifies to f⁻¹(4) = 43 / 10, or f⁻¹(4) = 4.3. Therefore, the value of the inverse function at x = 4 is 4.3. This step completes the process of finding f⁻¹(4) for the given function. The entire process involved substitution, algebraic manipulation, and a clear understanding of how to find the inverse of a function.

(c) If f(4x + 5) = 20x + 31, find f⁻¹(4)

In this section, we aim to determine f⁻¹(4), given the function f(4x + 5) = 20x + 31. This problem, similar to the previous one, requires us to find the inverse function by expressing x in terms of f(4x + 5). The steps involved are crucial for understanding how to manipulate functions and find their inverses, a fundamental concept in mathematics. This detailed explanation will guide you through each step, making the process clear and understandable.

First, we introduce a substitution to simplify the function. Let y = 4x + 5. Substituting y into the equation f(4x + 5) = 20x + 31, we get f(y) = 20x + 31. Now, we need to express x in terms of y. From the substitution y = 4x + 5, we isolate x by first subtracting 5 from both sides, yielding y - 5 = 4x. Then, we divide both sides by 4 to get x = (y - 5) / 4. Substitute this expression for x back into the equation f(y) = 20x + 31. This gives us f(y) = 20((y - 5) / 4) + 31.

Simplifying this expression, we first divide 20 by 4, which gives us 5. So, f(y) = 5(y - 5) + 31. Expanding this, we get f(y) = 5y - 25 + 31, which simplifies to f(y) = 5y + 6. Now that we have f(y) expressed in terms of y, we can proceed to find the inverse function. To find the inverse function, we swap x and y and solve for y. So, let's rewrite f(y) = 5y + 6 as x = 5y + 6. Next, we solve for y. Subtract 6 from both sides to get x - 6 = 5y. Then, divide both sides by 5 to isolate y, resulting in y = (x - 6) / 5. This expression for y represents the inverse function, which we denote as f⁻¹(x) = (x - 6) / 5. We have successfully determined the inverse function.

Finally, to find f⁻¹(4), we substitute x = 4 into the inverse function we just derived. So, f⁻¹(4) = (4 - 6) / 5. This simplifies to f⁻¹(4) = -2 / 5, or f⁻¹(4) = -0.4. Therefore, the value of the inverse function at x = 4 is -0.4. This step concludes the process of finding f⁻¹(4) for the given function. The entire process involved substitution, algebraic manipulation, and a clear understanding of how to find the inverse of a function.

Conclusion

In conclusion, finding the inverse function value, specifically f⁻¹(4), for given functions involves a systematic approach that includes substitution, algebraic manipulation, and a clear understanding of function inverses. By substituting and simplifying the given functional equations, we can express x in terms of y, which allows us to derive the inverse function. Once the inverse function is found, evaluating it at a specific point, such as x = 4, is a straightforward process of substitution. These skills are fundamental in mathematics and are essential for solving more complex problems in calculus and beyond. The detailed explanations and step-by-step solutions provided in this guide offer a comprehensive understanding of the process, ensuring clarity and comprehension for anyone studying inverse functions.