Problem 15 - College Algebra

Question

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x)) .\) Simplify your answers. $$f(x)=\sqrt[3]{x}, g(x)=\frac{x+1}{x^{3}}$$

Step-by-Step Solution

Answer

\(f(g(x)) = \frac{(x+1)^{1/3}}{x}, g(f(x)) = \frac{\sqrt[3]{x} + 1}{x}.\)

Step 1: Find the Function Composition for f(g(x))

We need to substitute the function \(g(x)\) into \(f(x)\). Start by substituting \(g(x)\) as the input for \(f(x)\):\[ f(g(x)) = \sqrt[3]{g(x)} = \sqrt[3]{\frac{x+1}{x^3}}. \] So, the expression for \(f(g(x))\) is \(\sqrt[3]{\frac{x+1}{x^3}}\).

Step 2: Simplify f(g(x))

The expression \(\sqrt[3]{\frac{x+1}{x^3}}\) can be simplified by taking the cube root of the numerator and the denominator separately:\[ f(g(x)) = \frac{(x+1)^{1/3}}{x}. \] This is the simplified form of \(f(g(x))\).

Step 3: Find the Function Composition for g(f(x))

Now, we need to substitute the function \(f(x)\) into \(g(x)\). Start by substituting \(f(x)\) as the input for \(g(x)\):\[ g(f(x)) = g(\sqrt[3]{x}) = \frac{\sqrt[3]{x} + 1}{(\sqrt[3]{x})^3}. \]This simplifies to \[ g(f(x)) = \frac{\sqrt[3]{x} + 1}{x}. \]

Step 4: Simplify g(f(x))

The expression \(\frac{\sqrt[3]{x} + 1}{x}\) is already in its simplest form. The expression cannot be simplified further in terms of elementary functions. Therefore, \(g(f(x)) = \frac{\sqrt[3]{x} + 1}{x}.\)

Key Concepts

Understanding Algebraic Function Composition

Function composition in algebra involves taking two different functions and creating a new function by plugging one function into another. This means replacing the variable of the first function with the entire expression of the second. Consider two functions \( f(x) \) and \( g(x) \). The function composition of \( f \) and \( g \) is denoted as \( f(g(x)) \).

• Input Replacement: When you perform the composition \( f(g(x)) \), you substitute \( g(x) \) into \( f(x) \) as the new input.
• Order Matters: The order of composition is important. \( f(g(x)) \) is generally not equivalent to \( g(f(x)) \).

In our exercise, we find \( f(g(x)) = \sqrt[3]{\frac{x+1}{x^{3}}} \). Here, the arguments of \( f \) and \( g \) are replaced and simplified as needed to create a new combined function.

Simplifying Expressions in Composition

Simplifying expressions is a crucial skill in algebra when dealing with composed functions. The goal is to make expressions easier to work with and understand. By reducing complex expressions, better insights into their behavior and properties are gained. In function composition, simplification may involve:

• Breaking Down Fractions: For instance, \( \sqrt[3]{\frac{x+1}{x^{3}}} \) allows for independent simplification by evaluating the cube root of the numerator \( x+1 \) and the denominator \( x^{3} \).
• Applying Factorization: Whenever possible, restructure the expression into a simpler form. For \( f(g(x)) = \frac{(x+1)^{1/3}}{x} \), recognizing the simpler forms helps in further computations.

Simplifying expressions is a necessary step in solving equations, understanding the domain of the function, and graph behavior.

Inverse Functions and Their Role in Algebra

Inverse functions reverse the process enacted by the original function. If you have a function \( f(x) \), its inverse function, denoted \( f^{-1}(x) \), will return the original input when applied:

• Process Reversal: If \( y = f(x) \), then \( x = f^{-1}(y) \).
• Graphical Representation: The graph of a function and its inverse are reflections across the line \( y=x \).

Finding the inverse can help solve equations where swapping outputs with inputs simplifies solving for a variable. In combination with function composition, inverse functions can offer insights into solving systems of equations or discovering unknown functions. Although our exercise doesn't directly involve finding an inverse, understanding their mechanism offers a broader comprehension of algebraic function properties and their manipulation.