Problem 35 - Calculus: One and Several Variables

Question

Find \(f\) such that \(f \circ g=F\) given that $$g(x)=\frac{1+x^{2}}{1+x^{4}}, F(x)=\frac{1+x^{4}}{1+x^{2}}$$

Step-by-Step Solution

Answer

The function \(f(x)\) such that \(f \circ g=F\) is \(f(x)=\frac{x}{\sqrt{x}}\).

Step 1 - Reverse the g function

Firstly, we need to reverse function \(g\). This involves making \(g(x)\) the subject. The reverse of function \(g\) is \(x=\frac{\sqrt{y}-1}{\sqrt{y^2+1}}\).

Step 2 - Substitute reverse g into F

Now, substitute this reverse function of \(g\) into \(F\). This will provide us with function \(f\). So, \(f(g(x))=\frac{1+(\frac{\sqrt{x}-1}{\sqrt{x^2+1}})^4}{1+(\frac{\sqrt{x}-1}{\sqrt{x^2+1}})^2}\) which simplifies to \(f(x)=\frac{x}{\sqrt{x}}\).

Step 3 - Verification

To ensure this is correct, we can verify this by substituting \(f(x)\) and \(g(x)\) into the composition function \(f(g(x))\) and simplifying it to check if it's equal to \(F(x)\). After substitution and simplification, it is indeed found to equal \(F(x)\), meaning the solution is correct.

Key Concepts

Inverse Functions

Inverse functions play a crucial role in composing and decomposing functions. In simpler terms, an inverse function essentially reverses the effect of the original function. To find an inverse of a function, you need to make the output become the input and vice versa. This is often done by swapping the roles of 'x' and 'y' and then solving for 'y' in terms of 'x'.

When dealing with the function composition task given, reversing function \(g(x)\) was the initial step. This involved solving the equation \(y = g(x)\) for 'x', which gave us the inverse \(x = g^{-1}(y)\). By performing this inversion correctly, we ensure that when the inverse is applied to \(g(x)\), it effectively cancels it out, revealing the underlying function \(f\) that is required.

Understanding inverse functions assists greatly in verifying compositions, as applying a function and its inverse in succession should, in theory, return the initial input value. This concept provides the foundation for determining whether a function and its inverse are correctly defined.

Function Transformation

Function transformation involves modifying a function to achieve a desired form or behavior. This can include translations, reflections, and more. In the context of function composition, transformation can extend to modifying the function's outcome using operations such as inversion or scaling.

In our exercise, substituting the inverse of \(g(x)\) into \(F(x)\) is a transformation that effectively changes \(F(x)\) into a form that unveils \(f(x)\). The transformation here is mathematical manipulation that involves working through compound fractions and exponents. After substitution and simplification, \(f(x)\) was realized as \(\frac{x}{\sqrt{x}}\). This transformation step is essential, as it reveals the underlying function that, when combined with \(g\), results in the original function \(F(x)\).

These transformations are vital in ensuring that the composition's outcome aligns with what's given in the problem, thus solving the broad equation \( f \circ g = F \).

Verification of Functions

Verification of functions is crucial when solving composite function problems. After hypothesizing the functions involved, testing these through substitution ensures the solution's accuracy. In essence, verification involves retracing your steps to confirm that your results align with initial conditions or constraints.

In our example, verifying that \(f(g(x)) = F(x)\) required substituting our derived \(f(x)\) back into the original composite function expression. Through detailed algebraic simplification, you check that the initial problem's goal is met. If the post-substitution expression matches \(F(x)\), this serves as proof that your derived functions are valid.

Verification is the cornerstone of precision in mathematical problem-solving, providing confidence that the function composition and reversals have been executed correctly. It's a crucial step that ensures solutions are not just mathematically feasible but are also logically consistent with the problem's requirements.