Problem 39 - Calculus

Question

Determine functions \(f\) and \(g\) such that \(h(x)=f(g(x)) .\) [Note: There is more than one correct answer. Do not choose \(f(x)=x \text { or } g(x)=x\).] $$h(x)=e^{2 x}$$

Step-by-Step Solution

Answer

Choose \( f(x) = e^x \) and \( g(x) = 2x \).

Step 1: Understand the Function Composition

The function composition problem asks us to express the given function \( h(x) = e^{2x} \) as a composition of two functions \( f \) and \( g \), such that \( h(x) = f(g(x)) \).

Step 2: Choose an Inner Function \( g(x) \)

One possible choice for \( g(x) \) is to take a function related to the structure of \( e^{2x} \). We can choose \( g(x) = 2x \) so that the exponent of \( e \) is simplified.

Step 3: Determine the Outer Function \( f(x) \)

With \( g(x) = 2x \), we need \( f(x) \) such that \( f(g(x)) = f(2x) = e^{2x} \). A suitable outer function would be \( f(x) = e^x \), ensuring \( f(2x) = e^{2x} \).

Step 4: Verify the Composition

Substitute \( g(x) = 2x \) into \( f(x) = e^x \). This gives \( f(g(x)) = f(2x) = e^{2x} \), which matches the original function \( h(x) = e^{2x} \).

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the context of the exercise, our exponential function is given by \( h(x) = e^{2x} \). Here, \( e \) is Euler's number, approximately equal to 2.718, a constant that forms the base of natural logarithms. When you see an expression like \( e^{2x} \), it indicates exponential growth (or decay, if the exponent is negative). Exponential functions are ubiquitous in science, especially in biology for modeling populations, physics for radioactive decay, and finance for calculating compound interest. Understanding these functions' behavior is crucial for solving problems that involve rapid growth or decay.

Inner Function

The inner function is the first function you apply when performing function composition. It's a crucial piece of the puzzle as it sets the stage for the outer function. In this exercise, we identified the inner function \( g(x) = 2x \). This choice was made because it simplifies the exponent within the exponential function. The purpose of the inner function is to transform the input in a way that when you apply the outer function to it, you recreate the original function. In general, choosing a wise inner function makes it easier to find a suitable outer function. It can be any function that, once composed with another, results in the desired function.

Outer Function

The outer function is applied last in function composition, and it acts on the result of the inner function. For the given composition problem, the outer function we chose is \( f(x) = e^x \). The idea here is that by applying \( f(x) \) to the result of the inner function \( g(x) \), you achieve the overall behavior of the original function \( h(x) \). It is important that when applied to the output of the inner function, the outer function should recreate the original function \( h(x) = e^{2x} \). This way, function composition transforms an existing function into a product of two simpler functions.

Function Decomposition

Function decomposition involves breaking down a complex function into compositions of simpler functions. The goal is to simplify analysis or computations. The original function \( h(x) = e^{2x} \) was decomposed into two simpler functions: \( f(x) = e^x \) and \( g(x) = 2x \). The process requires a strategic choice of functions so that when they are composed, they yield the original complex function. Decomposition is a powerful tool not only in algebra but also in fields like programming and systems design, where complex tasks are broken into manageable components. Understanding function decomposition provides a deeper insight into function structures and how complex behaviors can be synthesized from simpler elements.