Problem 32 - Algebra and Trigonometry

Question

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x^{3}+2, \quad g(x)=\sqrt[3]{x} $$

Step-by-Step Solution

Answer

\( f \circ g(x) = x + 2, g \circ f(x) = \sqrt[3]{x^3 + 2}, f \circ f(x) = (x^3 + 2)^3 + 2, g \circ g(x) = x^{1/9} \); all domains are all real numbers.

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

To find \( f \circ g \), substitute \( g(x) = \sqrt[3]{x} \) into \( f(x) = x^3 + 2 \). This results in: \( f(g(x)) = (\sqrt[3]{x})^3 + 2 = x + 2 \). The domain of \( g(x) = \sqrt[3]{x} \) is all real numbers, so the domain of \( f \circ g \) is also all real numbers.

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

To find \( g \circ f \), substitute \( f(x) = x^3 + 2 \) into \( g(x) = \sqrt[3]{x} \). This results in: \( g(f(x)) = \sqrt[3]{x^3 + 2} \). The expression \( x^3 + 2 \) is always real, so the domain of \( g \circ f \) is all real numbers.

Step 3: Compute \( f \circ f \)

To find \( f \circ f \), substitute \( f(x) = x^3 + 2 \) into itself. This results in: \( f(f(x)) = (x^3 + 2)^3 + 2 \). The domain of \( f(x) = x^3 + 2 \) is all real numbers, so the domain of \( f \circ f \) is also all real numbers.

Step 4: Compute \( g \circ g \)

To find \( g \circ g \), substitute \( g(x) = \sqrt[3]{x} \) into itself. This results in: \( g(g(x)) = \sqrt[3]{\sqrt[3]{x}} = x^{1/9} \). The domain of \( g(x) = \sqrt[3]{x} \) is all real numbers, so the domain of \( g \circ g \) is also all real numbers.

Key Concepts

Understanding the Domain of a Function

The domain of a function is a crucial concept, as it defines the set of input values for which the function is defined. To determine a function's domain, we consider all values of "x" that can be plugged into the function without causing any mathematical errors. For the given functions:

• For a cubic function like \(f(x) = x^3 + 2\), the domain includes all real numbers, because you can cube any real number and add 2 to it without any issues.
• Similarly, for a cube root function \(g(x) = \sqrt[3]{x}\), the domain is also all real numbers. Unlike a square root, which requires non-negative numbers to avoid imaginary results, cube roots have no such restrictions. You can take the cube root of any real number.

Both functions in this exercise have domains that include all real numbers. This means there isn't a single "bad" input value that would make any function undefined. Behind this simplicity lies the beauty and versatility of these mathematical expressions.

Exploring Cubic Functions

Cubic functions are polynomial functions with the highest exponent of "3". They take the form \(f(x) = ax^3 + bx^2 + cx + d\). Our function, \(f(x) = x^3 + 2\), simplifies the typical structure by having "b", "c", and "d" equal to zero and two, respectively. Here’s what makes cubic functions special:

• The graph of a cubic function typically resembles the letter 'S', with twists due to its turning points, forming a smooth curve known as an inflection point.
• Cubic functions can have up to three real roots, meaning they can cross the x-axis up to three times.
• These functions not only extend indefinitely in both directions but are also continuous and smooth across their entire domain.

Understanding these properties is essential as they influence the behavior of functions like \(f \circ f(x)\), where a function is composed with itself, leading to new but predictable patterns on a graph. Cubic functions pave the way for understanding more complex polynomial functions.

Demystifying the Cube Root

The cube root is the inverse operation of cubing a number. When we talk about \(g(x) = \sqrt[3]{x}\), we're referring to the value that, when cubed, returns "x". Here’s some vital information:

• A cube root can handle any real number as input, meaning the domain is all real numbers. This is because even negative numbers have real cube roots.
• Unlike the square root, which produces two possible roots (positive and negative), the cube root is unique and consistent in providing one real valued answer.
• Cube roots help reverse the cubing process, which can be an intricate tool in solving equations or simplifying composite functions like \(g \circ g(x)\).

Engaging with cube roots gives you the ability to explore deeper into function operations, especially in understanding inverses and finding simplified forms for complex solutions.