- The problem requires decomposing \(H(x) = 4x^6 + 3\) into two functions \(f(x)\) and \(g(x)\) such that \(f(g(x)) = H(x)\), with neither being the identity function.
- By choosing the inner function \(g(x) = x^3\), we can determine the corresponding outer function \(f(u) = 4u^2 + 3\).
- Verifying the functions against the identity function requirement: \(g(x) = x^3
eq x\) and \(f(x) = 4x^2 + 3
eq x\).
- Final answer is the pair of functions \(f(x) = 4x^2 + 3\) and \(g(x) = x^3\). \(\boxed{f(x) = 4x^2 + 3, g(x) = x^3}\)
Explanation
- Understanding the Problem and Decomposition
We are given the composite function \(H(x) = 4x^6 + 3\) and need to find two non-identity functions \(f(x)\) and \(g(x)\) such that \((f \circ g)(x) = f(g(x)) = 4x^6 + 3\). An identity function is defined as \(I(x) = x\). By decomposing the expression \(4x^6 + 3\), we can identify layers of operations that correspond to the outer function \(f(u)\) and the inner function \(g(x)\).
- Finding a Valid Pair of Functions
One straightforward decomposition is to treat the power part as the inner function. Let the inner function be \(g(x) = x^3\). Then we can substitute \(g(x)\) into the original expression:
\[H(x) = 4(x^3)^2 + 3 = 4(g(x))^2 + 3.\]
This suggests that the outer function \(f(u)\) is defined as \(f(u) = 4u^2 + 3\).
We check if either function is the identity function:
- \(g(x) = x^3
eq x\), so it is not the identity function.
- \(f(u) = 4u^2 + 3
eq u\), so it is not the identity function.
Thus, the pair of functions \(f(x) = 4x^2 + 3\) and \(g(x) = x^3\) is a valid solution.
- Conclusion
We have identified a valid pair of functions \(f(x) = 4x^2 + 3\) and \(g(x) = x^3\). Since the problem states there may be more than one correct answer, we have confirmed that this choice satisfies all conditions provided, including the non-identity requirement. The composition is \(f(g(x)) = f(x^3) = 4(x^3)^2 + 3 = 4x^6 + 3 = H(x)\). The final answer is the pair of functions found.
Examples
In computer programming, function composition is a powerful tool where you combine simple, reusable blocks of code to perform complex operations. For example, consider a process where you first scale a set of data (like converting units) and then apply a transformation (like squaring for energy calculations). By breaking a complex equation \(H(x)\) into smaller functions \(f(x)\) and \(g(x)\), you are essentially modularizing your algorithm. This makes code easier to test, debug, and reuse across different parts of a software project.