Problem 83 - College Algebra

Question

Use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). What is the domain of \((f \circ g)(x) ?\)

Step-by-Step Solution

Answer

The domain of \\( (f \circ g)(x) \\) is all real numbers or \\ (-\infty, \infty) \\.

Step 1: Understand the Composition

The function composition \(f \circ g\left(x\right)\) is defined as \((f \circ g)(x) = f(g(x))\). So, we need to understand the behavior of \(! g(x) = \sqrt[3]{x-1} \) first.

Step 2: Determine the Domain of g(x)

The function \(g(x) = \sqrt[3]{x-1}\) represents a cube root function. Cube root functions are defined for all real numbers, meaning the domain of \(g(x)\) is \(-\infty, \infty\).

Step 3: Determine the Domain of f(x)

The function \(f(x) = x^3 + 1\) is a polynomial function. Polynomial functions are defined for all real numbers, so the domain of \(f(x)\) is also \(-\infty, \infty\).

Step 4: Determine the Domain of (f ∘ g)(x)

Since \(g(x)\) is defined for all real numbers, and \(f(x)\) is defined wherever \(g(x)\) is defined, the domain of \(f \circ g(x)\) is all real numbers. Hence, the domain is \(-\infty, \infty\).

Key Concepts

Function Composition

Function composition involves combining two functions into a single function. The notation \(f \circ g\left(x\right)\) denotes the function composition of \(f\) and \(g\), which means "apply \(g\) and then \(f\)". In simpler terms, you first apply the function \(g(x)\), and then use the result as the input for the function \(f(x)\).

If you imagine function \(g(x)\) as a machine that transforms inputs, then \(f(x)\) is the machine that processes \(g(x)\)'s outputs. This results in a new function: \(f(g(x))\).

Function composition is a crucial concept in mathematics as it builds new functions from existing ones, enabling more complex operations.

Always ensure that the domain of \(g(x)\) covers the range sufficiently so that \(f\) can accept every result \(g\) produces.

Cube Root Function

The cube root function, denoted as \(\sqrt[3]{x}\), allows us to find a number which, when multiplied by itself twice, equals \(x\). Unlike square roots, cube roots are defined for all real numbers. This is because any real number, negative or positive, has a real cube root.

For the cube root function \(g(x) = \sqrt[3]{x - 1}\), this translates to shifting the basic cube root function to the right by 1 unit. This shift results because \(-1\) is inside the function with \(x\).

In operations, cube roots provide flexibility in the domain because they can handle negative and positive values, unlike even-degree roots, which are limited to non-negative numbers.

Polynomial Function

A polynomial function is an expression consisting of variables, coefficients, and non-negative integer exponents. Functions like \(f(x) = x^3 + 1\) are common examples. Here, the highest power of \(x\) is 3, making it a cubic polynomial.

Key characteristics of polynomial functions include:

• They are continuous, meaning they have no breaks, holes, or gaps in their graphs.
• They are smooth, meaning their graphs display gentle, sweeping curves.
• The domain of polynomial functions encompasses all real numbers, \((-\infty, \infty)\), because you can substitute any real number into the expression without restrictions.

Polynomials form the backbone of many mathematical models and tools due to these versatile properties.

Real Numbers

Real numbers include all the numbers we commonly use in mathematics, both rational numbers like \(3, -2, \frac{1}{4}\), and irrational numbers like \(\pi, \sqrt{2}\).

Represented by \(\mathbb{R}\), the set of real numbers is vital because it forms the base set from which many functions draw their domains.

Real numbers have the following properties:

• They can be plotted on a number line, spanning from negative infinity to positive infinity without breaks.
• They fill "real-world" quantities like length, weight, and temperature due to their continuity and inclusivity.

The function domains we explored, such as the cube root and polynomial functions, are defined over all real numbers, showcasing the reach and importance of this comprehensive numerical set.