Problem 3 - Applied Calculus

Question

In Problems \(1-3,\) find the following: (a) \(\quad f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=3 x \text { and } g(x)=e^{2 x}$$

Step-by-Step Solution

Answer

(a) \(f(g(x)) = 3e^{2x}\); (b) \(g(f(x)) = e^{6x}\); (c) \(f(f(x)) = 9x\).

Step 1: Determine f(g(x))

To find \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). Since \(g(x) = e^{2x}\), we replace all \(x\) in \(f(x)\) with \(e^{2x}\):\[f(g(x)) = f(e^{2x}) = 3 imes e^{2x} = 3e^{2x}.\]

Step 2: Determine g(f(x))

To find \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). Since \(f(x) = 3x\), replace all \(x\) in \(g(x)\) with \(3x\):\[g(f(x)) = g(3x) = e^{2(3x)} = e^{6x}.\]

Step 3: Determine f(f(x))

To find \(f(f(x))\), substitute \(f(x)\) into itself. Since \(f(x) = 3x\), replace all \(x\) in \(f(x)\) with \(3x\):\[f(f(x)) = f(3x) = 3(3x) = 9x.\]

Key Concepts

Composite Functions

When we talk about composite functions, we deal with situations where one function is inputted into another function. Imagine you're following a two-step process: you take the result from the first function and plug it into the second one.

A common notation for composite functions uses the symbols of the functions involved. For instance, if we have two functions, say \(f(x)\) and \(g(x)\), their composition might be written as \(f(g(x))\) or \(g(f(x))\). The expression \(f(g(x))\) means you first apply \(g(x)\) to your input \(x\), and then you use the result as an input to \(f(x)\).

Remember when composing functions:

• Order matters: \(f(g(x))\) is generally not the same as \(g(f(x))\).
• Think of \(g(x)\) as the inside function and \(f(x)\) as the outside function when you are solving \(f(g(x))\).

Composite functions are useful in mathematics because they allow more complex processes to be broken down into manageable steps.

Exponential Functions

Exponential functions are a specific class of functions where the variable appears in the exponent. A standard form is \(f(x) = a^{x}\), where \(a\) is a constant base. In this context, we see \(g(x) = e^{2x}\), which is an exponential function with base \(e\), the natural exponent.

The function \(g(x) = e^{2x}\):

• Grows rapidly as \(x\) increases due to the multiplying effect of the exponent.
• Is used in many real-world applications like compound interest and natural processes.

Understanding exponential functions is important because they model phenomena where growth or decay accelerates over time. In calculus, they have unique properties that help solve various types of problems.

Substitution in Functions

Substitution in functions is a core operation in mathematics where one expression is replaced by another within a function. It allows us to understand how functions interact and how they might behave when composed.

To substitute successfully:

• Identify what part of the function is being replaced. For example, substituting \(g(x) = e^{2x}\) into \(f(x) = 3x\), you replace every \(x\) in \(f(x)\) with \(e^{2x}\), resulting in \(3e^{2x}\).
• When doing \(f(f(x))\), substitute \(f(x)\) back into itself, such as turning \(f(x) = 3x\) into \(f(3x) = 9x\).

Substitution simplifies complex expressions and is foundational for calculus and solving equations. It can also aid in creating models that better reflect real-world scenarios by adjusting variables dynamically.