Problem 39 - Precalculus with Limits

Question

In Exercises 37-40, find (a) \(f \circ g\), (b) \(g \circ f\), and (c) \(g \circ g\). \(f(x) = \sqrt[3]{x-1}\), \(g(x) = x^3 + 1\)

Step-by-Step Solution

Answer

The compositions are: (a) \(f \circ g = \sqrt[3]{x^3}\), (b) \(g \circ f = x\), and (c) \(g \circ g = x^9 + 3x^6 + 3x^3 + 2\)

Step 1: Find \(f \circ g\)

To calculate this, we need to replace every x in \(f(x)\) with \(g(x)\). So, \(f(g(x)) = \sqrt[3]{g(x) - 1} = \sqrt[3]{x^3 + 1 - 1} = \sqrt[3]{x^3}\

Step 2: Find \(g \circ f\)

Like before, replace every \(x\) in \(g(x)\) with \(f(x)\). So, \(g(f(x)) = f(x)^3 + 1 = (\sqrt[3]{x-1})^3 + 1 = x - 1 + 1 = x\

Step 3: Find \(g \circ g\)

Here, replace every x in \(g(x)\) with \(g(x)\). So, \(g(g(x)) = g(x)^3 + 1 = (x^3 + 1)^3 + 1 = x^9 + 3x^6 + 3x^3 + 2\

Key Concepts

Function Operations

Function operations involve combining two functions in a variety of mathematical ways. These combinations are important because they allow us to simplify complex expressions and explore relationships between different functions. There are several types of operations we can perform on functions, including addition, subtraction, multiplication, division, and composition.

Composition of functions, denoted as \(f \circ g\), is particularly interesting because it involves applying one function to the results of another function. Here’s how it works step by step:

• Take an input \(x\).
• Find \(g(x)\), where \(g\) is the first function applied.
• Then, take the result of \(g(x)\) as input for \(f(x)\).

This process essentially "nests" the second function inside the first, which can yield surprising and useful results. Understanding these operations helps build problem-solving skills, as they are applied extensively in calculus and higher-level mathematics.

Cubic Functions

Cubic functions are polynomial functions of degree three, typically written in the form \(f(x) = ax^3 + bx^2 + cx + d\), where \(a eq 0\). They are important in algebra because they can model a variety of real-world problems, such as volume calculations and motion under constant acceleration. Here are some key characteristics:

• The graph of a cubic function is a smooth, continuous curve that can have one or two turning points, creating a shape similar to an elongated "S".
• Cubic functions can have up to three real roots, and their end behavior depends on the leading coefficient \(a\). If \(a > 0\), the graph rises to the right; if \(a < 0\), it falls to the right.

Inserting a cubic function into another through composition, as seen in the function operation for \(g(g(x))\), results in a much more complex higher-degree polynomial, demonstrating the power and intricacy possible through function composition.

Cube Root Functions

Cube root functions are the inverses of cubic functions and are expressed as \(f(x) = \sqrt[3]{x-c} + d\). These functions are useful for solving equations where the variable is cubed, allowing you to "undo" the power of three. Here are some properties of cube root functions:

• The graph of a cube root function is the mirror image of its corresponding cubic function along the line \(y = x\).
• Unlike square root functions, cube root functions do not have domain restrictions because any real number can be cubed and have a real cube root.

In the exercise problem, the function \(f(x) = \sqrt[3]{x-1}\) demonstrates how substituting the results of a cubic function into a cube root function simplifies the operation, often resulting in straightforward solutions like the identity function \(x\), showing the inverse nature between cubic and cube root functions.