Problem 36 - Algebra and Trigonometry

Question

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

The functions are: \( f \circ g = x + 2 \), \( g \circ f = \sqrt[3]{x^3 + 2} \), \( f \circ f = (x^3 + 2)^3 + 2 \), \( g \circ g = x^{1/9} \), all with domain \( \mathbb{R} \).

Step 1: Understand Composition of Functions

The composition of functions is the process of applying one function to the results of another. For example, the composition \( f \circ g \) means we need to apply \( f(x) \) to \( g(x) \). Similarly, for \( g \circ f \), we apply \( g(x) \) to \( f(x) \).

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

The composition \( f \circ g \) is defined as \( f(g(x)) \). Substitute \( g(x) = \sqrt[3]{x} \) into \( f(x) = x^3 + 2 \). Thus, \( f(g(x)) = f(\sqrt[3]{x}) = (\sqrt[3]{x})^3 + 2 = x + 2 \). Since the cube root function and the polynomial can accept all real numbers, the domain of \( f \circ g \) is \( \mathbb{R} \).

Step 3: Determine \( g \circ f \)

The composition \( g \circ f \) is defined as \( g(f(x)) \). Substitute \( f(x) = x^3 + 2 \) into \( g(x) = \sqrt[3]{x} \). Thus, \( g(f(x)) = g(x^3 + 2) = \sqrt[3]{x^3 + 2} \). Since both \( f(x) \) and \( g(x) \) can accept all real numbers, the domain of \( g \circ f \) is \( \mathbb{R} \).

Step 4: Determine \( f \circ f \)

The composition \( f \circ f \) is defined as \( f(f(x)) \). Substitute \( f(x) = x^3 + 2 \) itself into \( f(x) \). Thus, \( f(f(x)) = f(x^3 + 2) = (x^3 + 2)^3 + 2 \). The function accepts all real numbers, so the domain is \( \mathbb{R} \).

Step 5: Determine \( g \circ g \)

The composition \( g \circ g \) is defined as \( g(g(x)) \). Substitute \( g(x) = \sqrt[3]{x} \) into itself. Thus, \( g(g(x)) = g(\sqrt[3]{x}) = \sqrt[3]{\sqrt[3]{x}} = x^{1/9} \). The domain is also \( \mathbb{R} \) as cube roots can accept all real numbers.

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values for which the function is defined. When you are asked to find the domain, you are essentially determining for which values of \(x\) the function works without any issues.

For example, the function \(f(x) = x^3 + 2\) is a polynomial, and polynomial functions are defined for all real numbers. Similarly, the function \(g(x) = \sqrt[3]{x}\) is a cube root function, which also accepts any real number as input.

• For \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), the domain remains the same as each function is fully capable of handling any real number input.

By understanding the domain of these individual functions, you can more easily determine the domains of their compositions.

Real Numbers

Real numbers are the set of all rational and irrational numbers. This means they include positive numbers, negative numbers, whole numbers, fractions, and even irrational numbers like \(\sqrt{2}\).

When considering functions like \(f(x) = x^3 + 2\) and \(g(x) = \sqrt[3]{x}\), it's important to note that the real number system allows any of these types of numbers as inputs.

• Because no restrictions are placed on the input, both functions can accept \(\mathbb{R}\), the mathematical symbol for real numbers, as their domain.

By understanding the characteristics of real numbers, you can appreciate why these functions are so versatile in their domains.

Cube Root Function

The cube root function, expressed as \(g(x) = \sqrt[3]{x}\), is a special type of root function that can work with any input from the set of real numbers. Unlike square roots, which require non-negative numbers to yield real results, cube roots can also take negative numbers and give a real number as output.

This property makes cube root functions extremely flexible. For example:

• \(\sqrt[3]{8} = 2\)
• \(\sqrt[3]{-8} = -2\)

Because of this flexibility, the cube root function retains \(\mathbb{R}\) as its domain, significantly simplifying the determination of domains in composite functions involving cube roots.

Polynomial Functions

Polynomial functions, such as \(f(x) = x^3 + 2\), consist of variables raised to whole number exponents and coefficients. One key feature of polynomial functions is their ability to accept all real numbers as input.

This is because polynomials are defined and continuous over the entire real number line.

• They have no asymptotes, holes, or breaks, making them applicable for any \(x\)-value you choose from the set \(\mathbb{R}\).

These characteristics ensure that whether you are working with simple or complex polynomial constructions, their domains generally cover all real numbers without exception.