Problem 55 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

a. \(f(g(x)) = \frac{-2x^3 + 2}{(x^3 + 1)^2}\); b. \(g(f(x)) = \frac{x^6 - 3x^5 + 3x^4 - x^3 - 1}{x^6 - 3x^5 + 3x^4 - x^3 + 1}\); c. \(f(f(x)) = x^4 - 2x^3 + x\)

Step 1: Determine f(g(x))

To find \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). Use the expression for \(g(x)\): \(g(x) = \frac{x^3 - 1}{x^3 + 1}\). Substitute this into \(f(x) = x^2 - x\):\[f\left(g(x)\right) = \left(\frac{x^3 - 1}{x^3 + 1}\right)^2 - \frac{x^3 - 1}{x^3 + 1}\]This expression simplifies to:\[f\left(g(x)\right) = \frac{(x^3 - 1)^2}{(x^3 + 1)^2} - \frac{x^3 - 1}{x^3 + 1}\]Combine the fractions into a single one by finding a common denominator, resulting in:\[f\left(g(x)\right) = \frac{(x^3 - 1)^2 - (x^3 - 1)(x^3 + 1)}{(x^3 + 1)^2}\]Simplify the numerator:\[(x^3 - 1)^2 = x^6 - 2x^3 + 1, \quad (x^3 - 1)(x^3 + 1) = x^6 - 1\]So the simplified expression for the numerator is:\[x^6 - 2x^3 + 1 - (x^6 - 1) = -2x^3 + 2\]The final expression for \(f(g(x))\) is:\[f(g(x)) = \frac{-2x^3 + 2}{(x^3 + 1)^2}\]

Step 2: Determine g(f(x))

To find \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). Use the expression for \(f(x)\): \(f(x) = x^2 - x\). Substitute this into \(g(x) = \frac{x^3 - 1}{x^3 + 1}\):\[g\left(f(x)\right) = \frac{(x^2 - x)^3 - 1}{(x^2 - x)^3 + 1}\]Expand \((x^2 - x)^3\):\[(x^2 - x)^3 = (x^2 - x)(x^2 - x)(x^2 - x)\]Calculate \((x^2 - x)^2\):\[(x^2 - x)^2 = x^4 - 2x^3 + x^2\]Now multiply by \((x^2 - x)\) to find \((x^2 - x)^3\):\[(x^4 - 2x^3 + x^2)(x^2 - x) = x^6 - 2x^5 + x^4 - x^5 + 2x^4 - x^3 = x^6 - 3x^5 + 3x^4 - x^3\]Thus, substituting back:\[g(f(x)) = \frac{x^6 - 3x^5 + 3x^4 - x^3 - 1}{x^6 - 3x^5 + 3x^4 - x^3 + 1}\]

Step 3: Determine f(f(x))

To find \(f(f(x))\), substitute \(f(x)\) into itself. Begin with \(f(x) = x^2 - x\) and substitute:\[f\left(f(x)\right) = f(x^2 - x)\]Substitute this into the expression for \(f(x)\):\[f\left(x^2 - x\right) = (x^2 - x)^2 - (x^2 - x)\]First expand \((x^2 - x)^2\):\[(x^2 - x)^2 = x^4 - 2x^3 + x^2\]Then substitute back:\[f\left(x^2 - x\right) = x^4 - 2x^3 + x^2 - (x^2 - x)\]Simplify this expression:\[f(f(x)) = x^4 - 2x^3 + x^2 - x^2 + x = x^4 - 2x^3 + x\]

Step 4: Solution Completion

Having found all the necessary compositions, we conclude:- \(f(g(x)) = \frac{-2x^3 + 2}{(x^3 + 1)^2}\)- \(g(f(x)) = \frac{x^6 - 3x^5 + 3x^4 - x^3 - 1}{x^6 - 3x^5 + 3x^4 - x^3 + 1}\)- \(f(f(x)) = x^4 - 2x^3 + x\)

Key Concepts

Rational Functions

Rational functions are types of functions represented by the ratio of two polynomial expressions. Essentially, they are the division of one polynomial by another. Because they involve ratios, it's important to pay attention to where the denominator equals zero since that determines where the function is undefined.
Consider the rational function in our original problem:

• The function is expressed as \( g(x) = \frac{x^3 - 1}{x^3 + 1} \).
• The numerator is \( x^3 - 1 \) and the denominator is \( x^3 + 1 \), both are polynomial expressions.

To understand rational functions, look at:

• Domain: It’s everywhere the denominator is not zero, so set \( x^3 + 1 eq 0 \).
• Asymptotic behavior: This happens if the polynomial expressions create a situation where the function trends towards a line, especially as \( x \) approaches infinity or points of undefined values.

Graphing these functions shows vertical and horizontal lines where the function doesn’t exist or approaches values closely. Rational functions are crucial for modeling scenarios in science and engineering, where relationships can be naturally expressed as ratios.

Polynomial Functions

Polynomial functions are foundational in mathematics and involve terms made up of variables raised to whole-number exponents. Their general structure can be simplified into the form: \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \) where the exponents are whole numbers and the coefficients are real numbers.
In the example given:

• \( f(x) = x^2 - x \) is a polynomial function.
• It is a simple quadratic polynomial, where \( x^2 \) represents the highest degree term.

Key attributes to remember about polynomial functions include:

• Degree: Defined by the highest power of \( x \), which guides the function’s shape.
• Roots: Points where the function equals zero, which can be solved algebraically.
• Behavior: With lower degree polynomials having curves, while higher degrees introduce more waves.

Polynomials are essential in calculus, physics, and engineering, as they simplify modeling real-world phenomena.

Function Operations

Function operations consist of various ways of combining functions, such as addition, subtraction, multiplication, division, and composition. Function composition, like in our exercise, allows you to apply one function to the results of another, combining their actions into a single operation.
Here's a summary of function operations applied in our steps:

• For \( f(g(x)) \), replace \( x \) in \( f(x) \) with \( g(x) \) producing \( f\left(\frac{x^3 - 1}{x^3 + 1}\right) \).
• For \( g(f(x)) \), replace \( x \) in \( g(x) \) with \( f(x) \) giving \( g(x^2 - x) \).
• For the self-composition \( f(f(x)) \), substitute \( f(x) \) into itself, resulting in \( f(x^2 - x) \).

Function composition transforms and combines functions, often leading to new functions with unique features. Every step essentially involves plugging one function into another, making it a versatile tool in advanced mathematics, modeling, and computer science. The critical point is to systematically replace and simplify terms to ensure accurate compositions.