Question

If \(f(x)=\frac{1}{1+x}\) and \(g(x)=\frac{1}{2+x}\), determine \(f(g(x))\).

Short answer

\(f(g(x)) = \frac{2+x}{3+x}\).

Step-by-step solution

- Understand the Functional Composition

To find the composite function, denoted as \(f(g(x))\), substitute the function \(g(x)\) into the function \(f(x)\).

- Identify \(g(x)\)

Given \(g(x) = \frac{1}{2+x}\), this is the function that needs to be substituted into \(f(x)\).

- Substitute \(g(x)\) into \(f(x)\)

Substitute the expression for \(g(x)\) into \(f(x)\). So we have: \[ f(g(x)) = f\left(\frac{1}{2+x}\right). \]

- Apply the Expression

Next, use the definition of \(f(x)\) in the above step. Since \(f(x) = \frac{1}{1+x}\), substitute \(\frac{1}{2+x}\) for \(x\) in \(f(x)\): \[ f\left(\frac{1}{2+x}\right) = \frac{1}{1 + \frac{1}{2+x}}. \]

- Simplify the Expression

Simplify the complex fraction. Begin by getting a common denominator in the denominator: \[ f\left(\frac{1}{2+x}\right) = \frac{1}{1 + \frac{1}{2+x}} = \frac{1}{\frac{(2+x) + 1}{2+x}} = \frac{1}{\frac{3+x}{2+x}}. \] This simplifies further to: \[ \frac{1}{\frac{3+x}{2+x}} = \frac{2+x}{3+x}. \]

Key concepts

Composite Functions

Composite functions are essential in understanding how one function can be applied to the results of another. This process is often denoted as \(f(g(x))\). To form a composite function, you substitute one function into another. In our given exercise, we start with \(f(x) = \frac{1}{1+x}\) and \(g(x) = \frac{1}{2+x}\). To find the composite function, \(f(g(x))\), we substitute \(g(x)\) into \(f(x)\). This results in \(f(g(x)) = f(\frac{1}{2+x} ) \), turning the problem into a fraction simplification.

Functional Notation

Functional notation is a way to represent functions mathematically, using symbols like \(f(x)\) or \(g(x)\). It allows us to understand how a function depends on its input. In our case, \(f(x) = \frac{1}{1+x}\) describes how the function \(f\) transforms \(x\). By replacing \(x\) with \(g(x)\), indicated by \(f(g(x))\), we trace how changes in the input affect the output through a sequence of operations. This makes functional notation crucial for complex problems involving multiple functions.

Fractions Simplification

Fraction simplification involves reducing complex expressions into simpler forms. In the given problem, we encounter \( \frac{1}{1 + \frac{1}{2+x}} \). Simplifying this fraction requires several steps:

• First, find a common denominator for the terms in the denominator.
• Rewrite the complex fraction.
• Simplify by canceling out terms.

After obtaining a common denominator, we arrive at \( \frac{1}{\frac{3+x}{2+x}} \). This simplifies further to \frac{2+x}{3+x} \, giving us a cleaner and more understandable result.

Precalculus

Mastering concepts like function composition, functional notation, and fractions simplification is essential in precalculus. These skills form the foundation for understanding more advanced mathematics. They help you manipulate and understand complex functions, paving the way for calculus and beyond. In our example, we tackled a composite function problem, reinforcing critical thinking and problem-solving skills crucial for your mathematical journey. Precalculus ensures that you are well-prepared for tackling calculus by solidifying your understanding of these fundamental concepts.