- Understand the definition of function composition: \(h(x) = f(g(x)) = \frac{1}{x^2+1}\) with \(g(x) = x^2+1\).
- Use substitution \(u = g(x) = x^2+1\) to transform the equation into \(f(u) = \frac{1}{u}\).
- Express the function in terms of \(x\) to get \(f(x) = \frac{1}{x}\).
- Verify the composition to confirm the result is \(\boxed{f(x) = \frac{1}{x}}\).
Explanation
- Analyzing the Function Composition
We are given the composition of two functions, \(h(x) = f(g(x))\), where \(h(x) = \frac{1}{x^2+1}\) and \(g(x) = x^2+1\). Our goal is to determine the expression for \(f(x)\). We know that \(f(g(x))\) means we substitute the expression for \(g(x)\) into \(f\). Thus, we are looking for a function \(f\) such that \(f(x^2+1) = \frac{1}{x^2+1}\).
- Solving for the Function \(f(x)\)
To find \(f(x)\), we can introduce a substitution variable \(u = g(x) = x^2+1\). Replacing \(x^2+1\) with \(u\) in the equation \(f(x^2+1) = \frac{1}{x^2+1}\), we obtain \(f(u) = \frac{1}{u}\). Since \(u\) represents any value that the function \(g(x) = x^2+1\) can take, and the domain of \(f\) must cover these values, we can replace the variable \(u\) with the standard input variable \(x\). Thus, we conclude that \(f(x) = \frac{1}{x}\).
- Verification and Conclusion
To verify our result, we check the composition \(f(g(x))\) using our determined \(f(x) = \frac{1}{x}\) and the given \(g(x) = x^2+1\):
\[f(g(x)) = f(x^2+1) = \frac{1}{x^2+1}.\]
This matches the given function \(h(x)\). Therefore, the function \(f(x)\) is indeed \(\frac{1}{x}\). Comparing this to the given options, we find that \(f(x) = \frac{1}{x}\) is the correct expression.
Examples
Function composition is like a factory assembly line where one machine processes raw material into an intermediate product, and the next machine processes that intermediate product into the final item. In mathematics, if machine \(g(x)\) transforms an input \(x\) into \(x^2+1\), and then machine \(f(u)\) takes that result and computes its reciprocal \(\frac{1}{u}\), the combined effect is the transformation \(x \to \frac{1}{x^2+1}\). This concept is fundamental in understanding how complex systems or algorithms can be broken down into simpler, modular steps, which is vital in fields ranging from computer programming to engineering and financial modeling.