Question

Determine the domain of (a) \(f\), (b) \(g\), and (c) \(f \circ g\). \(f(x)=\sqrt[3]{x+1}, \quad g(x)=x^{3}\)

Short answer

The domain of \(f(x)\), \(g(x)\), and \(f \circ g\) are all the set of all real numbers.

Step-by-step solution

Determine the domain of \(f(x)\)

The function \(f(x)=\sqrt[3]{x+1}\) is the cubic root of \(x+1\). The cubic root function is defined for all real numbers, so any \(x+1\) is valid. That means \(x\) can take any real number. Therefore, the domain for \(f\) is the set of all real numbers.

Determine the domain of \(g(x)\)

The function \(g(x)=x^{3}\) is a cubic function which is defined for all real numbers. So, the domain of \(g\) is also the set of all real numbers.

Determine the domain of \(f \circ g\)

To determine the domain of the composite function \(f \circ g\) we need to consider the domain of \(f\) after the function \(g\) has been applied. Start by substituting \(g(x)\) into \(f(x)\), which gives \(f(g(x))=f(x^{3})=\sqrt[3]{x^{3}+1}\). This function will also be defined for all real numbers because the cubic root function is defined for all real numbers. Therefore, the domain of the composite function is the set of all real numbers.

Key concepts

Cubic Root Function

A cubic root function is a type of function that involves the cube root of a variable expression. When you see a function like \( f(x) = \sqrt[3]{x+1} \), it means you take the cube root of the quantity \( x + 1 \). What's fascinating about the cubic root function is that it's defined for all real numbers. Unlike square roots, which are only defined for non-negative numbers, cube roots can handle negative inputs smoothly.
For example, if \( x = -8 \), the cubic root \( \sqrt[3]{-8} \) equals \(-2\), because \((-2)^3 = -8\).
Cubic root functions allow for a wide range of real number inputs:

• Small negative and positive values, like \(-0.1\) or \(0.1\), are easily managed without issue.
• Larger numbers, whether negative or positive, are similarly processed as the cube root is defined across the entire set of real numbers.

These characteristics make cubic root functions highly versatile and applicable in various mathematical contexts.

Composite Function

The concept of composite functions can seem complicated at first, but it's all about combining two functions to create a new one. When you see notation like \( f \circ g \), it represents the composition of two functions \( f \) and \( g \). Essentially, for any input \( x \), you apply \( g \) first, and then use the output of \( g(x) \) as the input for \( f \).
In our exercise, the composite function was created from \( f(x) = \sqrt[3]{x+1} \) and \( g(x) = x^3 \), resulting in \( f(g(x)) = \sqrt[3]{x^3 + 1} \). To find the domain of this composite function, we need to ensure that the result of \( g(x) \) will work in \( f(x) \).'

• Step 1: Compute \( g(x) = x^3 \). Since \( x^3 \) is defined for all real numbers, there's no restriction here.
• Step 2: Plug the result into \( f \), i.e., \( f(x^3) = \sqrt[3]{x^3 + 1} \). Since the cubic root function accepts all real numbers, \( f(g(x)) \) has no restrictions either.

This means the domain of the composite function \( f \circ g \) is also all real numbers, just like \( f \) and \( g \) individually.

Real Numbers

Real numbers are an essential concept in mathematics, forming the core set of numbers we frequently use in everyday calculations and advanced mathematical theories. This set includes both rational numbers (like fractions \( 1/2 \) or \(-4/3 \)) and irrational numbers (like \( \pi \) or \( \sqrt{2} \)).
When we talk about functions defined for all real numbers, it means every real number can be an input for the function without causing any undefined situations. For example, both the cubic root function \( \sqrt[3]{x+1} \) and the cubic function \( x^3 \) are defined for every possible real number value:

• Rational inputs are valid and will yield rational or irrational outputs.
• Irrational number inputs result in either irrational or special rational outputs, depending on the specific value.

The comprehensive nature of real numbers ensures that functions like those tackled in your exercise have broad and inclusive domains. You can be assured that functions involving real numbers typically operate seamlessly across this vast numerical landscape.