Problem 53 - Applied Calculus

Question

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=\sqrt{x}-1 ; g(x)=x^{3}-x^{2} $$

Step-by-Step Solution

Answer

a. \(f(g(x)) = \sqrt{x^3-x^2} - 1\); b. \(g(f(x)) = x^{3/2} + x^{1/2} - 2x + 1\); c. \(f(f(x)) = \sqrt{\sqrt{x} - 1} - 1\).

Step 1: Identify and note the given functions

We are given two functions: \( f(x) = \sqrt{x} - 1 \) and \( g(x) = x^3 - x^2 \). We will use these definitions to find the composite functions \( f(g(x)) \), \( g(f(x)) \), and \( f(f(x)) \).

Step 2: Find the composite function \( f(g(x)) \)

Substitute \( g(x) = x^3 - x^2 \) into the function \( f(x) \). Thus, \( f(g(x)) = f(x^3 - x^2) = \sqrt{x^3 - x^2} - 1 \). Therefore, \( f(g(x)) = \sqrt{x^3 - x^2} - 1 \).

Step 3: Find the composite function \( g(f(x)) \)

Substitute \( f(x) = \sqrt{x} - 1 \) into \( g(x) \). Hence, \( g(f(x)) = g(\sqrt{x} - 1) = (\sqrt{x} - 1)^3 - (\sqrt{x} - 1)^2 \). Expand these expressions to get the final form of \( g(f(x)) \).

Step 4: Expand \((\sqrt{x} - 1)^3\) and \((\sqrt{x} - 1)^2\)

First, expand \((\sqrt{x} - 1)^2 = x - 2\sqrt{x} + 1 \). Next, expand \((\sqrt{x} - 1)^3 = (\sqrt{x} - 1)(x - 2\sqrt{x} + 1) = x^{3/2} - 2x + x^{1/2} - \sqrt{x} + 2 - 1 = x^{3/2} + x^{1/2} - 3x + 2\).

Step 5: Complete \( g(f(x)) \) using expansions

Now substitute and simplify: \( g(f(x)) = x^{3/2} + x^{1/2} - 3x + 2 - (x - 2\sqrt{x} + 1) \). Simplifying gives \( g(f(x)) = x^{3/2} + x^{1/2} - 2x + 1 \).

Step 6: Find the composite function \( f(f(x)) \)

Substitute \( f(x) = \sqrt{x} - 1 \) back into itself: \( f(f(x)) = f(\sqrt{x} - 1) = \sqrt{\sqrt{x} - 1} - 1 \). Therefore, \( f(f(x)) = \sqrt{\sqrt{x} - 1} - 1 \).

Key Concepts

Function Composition

Function composition is an important concept in mathematics whereby you combine two or more functions to create a new function. This involves taking the output of one function and using it as the input for another function.
When we consider composite functions like \( f(g(x)) \), we're saying "input \( x \) into \( g \), and then take the result and input it into \( f \)." Let's look at this step by step:

• Identify Functions: First, know what your functions are. In our example, we have \( f(x) = \sqrt{x} - 1 \) and \( g(x) = x^3 - x^2 \). This sets the stage for composition.
• Substitution: For \( f(g(x)) \), we start by replacing \( x \) in \( f(x) \) with the entire function \( g(x) \), leading to \( \sqrt{x^3 - x^2} - 1 \).
• Calculate Results: The output \( \sqrt{x^3 - x^2} - 1 \) is the final composed function when you combine these two functions in the described order.

Function composition helps in transforming functions into more complex or simplified forms, broadening the scope of mathematical analysis.

Algebraic Functions

Algebraic functions involve operations like addition, subtraction, multiplication, division, and roots involving polynomial expressions. These functions often serve as the building blocks for more complex functions.
In the exercise, we worked with several algebraic functions:

• Polynomials: The function \( g(x) = x^3 - x^2 \) is a polynomial function. It uses powers of \( x \) and the typical algebraic operations of subtraction.
• Radical Functions: The function \( f(x) = \sqrt{x} - 1 \) is a radical function because it involves taking the square root.

Performing operations on algebraic functions, especially in the context of compositions like \( f(f(x)) \) which is \( \sqrt{\sqrt{x} - 1} - 1 \), requires careful manipulation of these expressions to ensure the transformations maintain validity within a realistic mathematical domain. It's essential to understand the nature of algebraic expressions to execute function operations correctly.

Mathematical Problem Solving

Problem-solving in mathematics involves applying a set of strategies to tackle complex tasks, such as composing functions. Here are some strategies highlighted in our step-by-step solution:

• Understand the Problem: Clearly define what you need to find. For example, the goal was to find \( f(g(x)) \), \( g(f(x)) \), and \( f(f(x)) \).
• Substitute and Simplify: Replace variables where needed and simplify expressions. This involves expanding expressions and reducing them to simpler forms.
• Verify and Revise: Check your work for errors or new insights. Make sure that each step logically follows from the previous one.

By adopting a structured approach to solving problems, especially in composing algebraic functions, mathematical problem solving becomes more accessible. Each step you tackle gives you more confidence in managing equations and function operations.