Problem 50 - Applied Calculus

Question

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=x^{8} ; \quad g(x)=2 x+5 $$

Step-by-Step Solution

Answer

a. \((2x + 5)^8\), b. \(2x^8 + 5\), c. \(x^{64}\)

Step 1: Substitute for f(g(x))

To find \(f(g(x))\), replace the \(x\) in \(f(x)=x^8\) with \(g(x)\). Since \(g(x) = 2x + 5\), substituting gives: \[f(g(x)) = (g(x))^8 = (2x + 5)^8\]

Step 2: Substitute for g(f(x))

To find \(g(f(x))\), replace the \(x\) in \(g(x)=2x + 5\) with \(f(x)\). Since \(f(x) = x^8\), substituting gives: \[g(f(x)) = 2(f(x)) + 5 = 2(x^8) + 5 = 2x^8 + 5\]

Step 3: Substitute for f(f(x))

To find \(f(f(x))\), replace the \(x\) in \(f(x)=x^8\) with \(f(x)\). Since \(f(x) = x^8\), substituting gives: \[f(f(x)) = (f(x))^8 = (x^8)^8 = x^{64}\]

Key Concepts

Understanding Function Evaluation

Function evaluation is the process of determining the value of a function given a specific input. This usually involves substituting the input value into the function's expression and simplifying to find the result.
For example, if you have a function like \( f(x) = x^8 \), evaluating \( f(2) \) means you'll substitute 2 for \( x \), resulting in \( 2^8 = 256 \).
It's like following a recipe: you have instructions and ingredients (the function and input), and you combine them to get your final dish (the result of the function).

• The goal is to understand how the function behaves uniquely for each input.
• Timely evaluations can help build intuition on how transformations affect the function's graph.

Overall, if you learn to manage function evaluations, you'll have a powerful tool for exploring mathematical relationships.

Exploring Nested Functions

Nested functions occur when one function is placed inside another function. This composition can be visualized as layers, where each layer adds complexity to the calculation.
Consider finding \( f(g(x)) \). Here, you are asked to input the result of \( g(x) \) into \( f(x) \).
If \( g(x) = 2x + 5 \) and \( f(x) = x^8 \), then calculating \( f(g(x)) \) involves substituting the output of \( g(x) \) into the input of \( f(x) \). Therefore:

• First, calculate \( g(x) = 2x + 5 \).
• Then substitute \( g(x) \) into \( f(x) \), resulting in \((2x + 5)^8\).

Nested functions are everywhere in calculus and advanced mathematics, illustrating complex relationships through simpler functions.

Mastering the Substitution Method

The substitution method is an effective technique for solving problems involving composite functions. It requires replacing a variable within a function’s definition with another expression.
This method is typically used in integration, differentiation, and composition problems. Following a structured process will make these problems easier.

• Identify which variable or expression will be substituted (e.g., replacing \( x \) with \( u \) or another function).
• Conduct the substitution systematically for clarity.
• Simplify the resulting expression, if possible.

In the exercises given, like finding \( f(f(x)) \) or \( g(f(x)) \), the substitution method helps us understand how one function significantly impacts another by changing its input. Using substitution, complex expressions are tackled one piece at a time, minimizing errors and enhancing comprehension.