Problem 55 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

a. Complex expression for f(g(x)); b. Complex expression for g(f(x)); c. \( f(f(x)) = x^4 - 2x^3 + x \).

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

To find \( f(g(x)) \), we need to substitute \( g(x) = \frac{x^3 - 1}{x^3 + 1} \) into \( f(x) = x^2 - x \). This makes \( f(g(x)) = \left( \frac{x^3 - 1}{x^3 + 1} \right)^2 - \frac{x^3 - 1}{x^3 + 1} \). Simplify the expression step-by-step.

Step 2: Square g(x)

Compute \( \left( \frac{x^3 - 1}{x^3 + 1} \right)^2 = \frac{(x^3 - 1)^2}{(x^3 + 1)^2} \). Expand and simplify to get \( \frac{x^6 - 2x^3 + 1}{x^6 + 2x^3 + 1} \).

Step 3: Simplify f(g(x)) expression

Substitute back into \( f(g(x)) = \frac{x^6 - 2x^3 + 1}{x^6 + 2x^3 + 1} - \frac{x^3 - 1}{x^3 + 1} \). Combine the fractions: \( \frac{(x^6 - 2x^3 + 1)(x^3 + 1) - (x^3 - 1)(x^6 + 2x^3 + 1)}{(x^6 + 2x^3 + 1)(x^3 + 1)} \). Simplify further if necessary.

Step 4: Substitute f(x) into g(x) for g(f(x))

To find \( g(f(x)) \), substitute \( f(x) = x^2 - x \) into \( g(x) = \frac{x^3 - 1}{x^3 + 1} \), leading to \( g(x^2 - x) = \frac{(x^2 - x)^3 - 1}{(x^2 - x)^3 + 1} \).

Step 5: Simplify g(f(x)) expression

Expand \( (x^2 - x)^3 \) to \( x^6 - 3x^5 + 3x^4 - x^3 \), then substitute to obtain \( g(f(x)) = \frac{x^6 - 3x^5 + 3x^4 - x^3 - 1}{x^6 - 3x^5 + 3x^4 - x^3 + 1} \). Simplify further as needed.

Step 6: Substitute f(x) into itself for f(f(x))

For \( f(f(x)) \), substitute \( f(x) = x^2 - x \) into the same function, resulting in \( f(f(x)) = ((x^2 - x)^2 - (x^2 - x)) \).

Step 7: Square f(x) inside f(f(x))

Expand \( (x^2 - x)^2 = x^4 - 2x^3 + x^2 \), substitute into \( f(f(x)) = x^4 - 2x^3 + x^2 - x^2 + x \). Simplify to get \( x^4 - 2x^3 + x \).

Key Concepts

Composite Functions

Composite functions occur when you take one function and apply it to the result of another function. Think of it as layering functions. For example, when you have two functions, such as \( f(x) \) and \( g(x) \), and you want to find \( f(g(x)) \), you're applying \( g(x) \) first, then taking that result and putting it into \( f(x) \).

The notation \( f(g(x)) \) means substitute every \( x \) in \( f(x) \) with the expression for \( g(x) \). Similarly, \( g(f(x)) \) means substitute every \( x \) in \( g(x) \) with the expression for \( f(x) \).

• When solving \( f(g(x)) \), start by calculating \( g(x) \) and then use that result in \( f(x) \).
• Always simplify your expressions step-by-step to help avoid errors.
• Composite functions are useful in many areas of math and science, as they allow complex operations to be broken down into simpler, sequential steps.

Understanding composite functions provides a powerful toolset for algebraic manipulation. This process is vital in many branches of math, including calculus and algebra.

Polynomials

Polynomials are mathematical expressions featuring sums of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable \( x \) is \( a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where \( a_n, a_{n-1}, ..., a_0 \) are constants and \( n \) is a non-negative integer.

• The degree of a polynomial is the highest power of the variable in the expression. For example, \( x^2 - x \) is a polynomial of degree 2.
• Polynomials are used in algebra to model and solve various equations and problems.
• Addition, subtraction, and multiplication of polynomials are straightforward. Division can be more complex and often requires methods like synthetic division or long division.

They form the backbone of more complex mathematical models like functions, including composite ones. For instance, in our example, \( f(x) = x^2 - x \) is a simple polynomial that can be easily manipulated to compose with another function.

Rational Functions

Rational functions are ratios of two polynomials. They are of the form \( \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not zero.

• These functions often have restrictions on the domain since the denominator cannot be zero. For example, in \( g(x) = \frac{x^3 - 1}{x^3 + 1} \), the denominator becomes zero when \( x^3 + 1 = 0 \), thus \( x eq -1 \).
• The behavior of rational functions can include vertical asymptotes, which occur where the denominator is zero, and horizontal asymptotes, determined by the leading coefficients of the polynomials.
• When simplifying rational functions, factor both the numerator and denominator and cancel any common factors, when appropriate, to minimize the expression.

Rational functions appear in various real-world applications, like physics and engineering. Understanding them allows you to predict how systems behave, especially when they approach certain critical points where the denominators tend to zero. In the context of composite functions, rational functions add a layer of complexity that requires careful manipulation and attention to their domains.