4. Find and interpret \( C=g(f(n)) \)

To address the problem **"4. Find and interpret \( C = g(f(n)) \)"**, we'll explore the concept of function composition and how it applies in this context. ### Understanding Function Composition **Function composition** involves applying one function to the result of another function. In mathematical terms, if you have two functions: - \( f: X \rightarrow Y \) - \( g: Y \rightarrow Z \) The composition \( g \circ f \) is a new function defined by: \[ (g \circ f)(n) = g(f(n)) \] This means you first apply \( f \) to \( n \), and then apply \( g \) to the result of \( f(n) \). ### Finding \( C = g(f(n)) \) Given the expression \( C = g(f(n)) \), here's how you can find \( C \): 1. **Apply Function \( f \) to \( n \):** - Calculate \( f(n) \). This gives you an intermediate result. 2. **Apply Function \( g \) to the Result of \( f(n) \):** - Take the value obtained from \( f(n) \) and apply \( g \) to it, resulting in \( g(f(n)) \). 3. **Assign the Result to \( C \):** - Set \( C = g(f(n)) \). **Example:** Let's consider specific functions to illustrate this process. - Let \( f(n) = 2n + 3 \) - Let \( g(m) = m^2 \) **Step 1: Apply \( f \) to \( n \)** \[ f(n) = 2n + 3 \] **Step 2: Apply \( g \) to \( f(n) \)** \[ g(f(n)) = g(2n + 3) = (2n + 3)^2 \] **Step 3: Assign to \( C \)** \[ C = (2n + 3)^2 \] ### Interpretation of \( C = g(f(n)) \) The composition \( C = g(f(n)) \) represents a two-step transformation of the variable \( n \): 1. **First Transformation (\( f \))**: The function \( f \) modifies \( n \) in a specific way (e.g., scaling and shifting). 2. **Second Transformation (\( g \))**: The function \( g \) takes the output of \( f(n) \) and transforms it further (e.g., squaring the result). The overall effect is a combined transformation where \( n \) undergoes both \( f \) and \( g \) sequentially. This can be useful in various applications, such as mathematical modeling, computer science algorithms, and more, where complex operations are built from simpler, composed functions. ### General Points to Consider - **Domain and Range**: Ensure that the range of \( f \) is compatible with the domain of \( g \) for the composition to be valid. - **Order Matters**: \( g(f(n)) \) is generally not the same as \( f(g(n)) \) unless the functions satisfy certain commutativity conditions. - **Simplification**: Sometimes, the composition can be simplified algebraically to a single expression, as shown in the example. If you have specific forms for \( f(n) \) and \( g(n) \), feel free to provide them, and I can offer a more detailed solution tailored to those functions!