Problem 83 - College Algebra

Question

For the following exercises, 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, \((-\infty, \infty)\).

Step 1: Understand the Composition Function

The composition function, denoted as \((f \circ g)(x)\), means \(f(g(x))\). In other words, you need to substitute \(g(x)\) into \(f(x)\).

Step 2: Identify the Domain for g(x)

The function \(g(x) = \sqrt[3]{x-1}\) is defined for all real numbers because cube roots exist for any real number. Therefore, the domain of \(g(x)\) is all real numbers, or \((-\infty, \infty)\).

Step 3: Determine f(g(x))

Substitute \(g(x)\) into \(f(x)\) to get \(f(g(x)) = (\sqrt[3]{x-1})^{3} + 1\). This simplifies to \((x-1) + 1 = x\).

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

Since \(f(g(x)) = x\), which is simply a linear function, it is defined for all real numbers. There are no restrictions, so the domain remains all real numbers, \((-\infty, \infty)\).

Key Concepts

Domain of a Function

In mathematics, the domain of a function is the complete set of possible input values (x-values) for which the function is defined. For the function composition \((f \circ g)(x)\), it is especially important to consider the domains of the individual functions involved. Knowing the domain helps us understand where a function is "safe" or valid to use. In our exercise, we were tasked with finding the domain of \((f \circ g)(x)\). The function \(g(x) = \sqrt[3]{x-1}\) is a cube root function, which means it is defined for all real numbers. This is because the cube root of any real number is also a real number. Thus, the domain of\(g(x)\) is all real numbers, or \((-\infty, \infty)\). Once we establish this, we move to the next step where \(f(x)=x^{3}+1\). The substitution \(g(x)\) into \(f(x)\) simplifies all down to \(x\), a linear function, which again is defined everywhere on the real line without restrictions. Thus, the domain of the composition \((f \circ g)(x)\) is the intersection of the domains of \(g(x)\) and \(f(x)\), reaffirming that the domain is indeed \((-\infty, \infty)\). This intuitive approach allows us to not only know where a function is defined, but also the extent of combining two functions.

Cube Root Function

The cube root function, denoted for our exercise as \(g(x) = \sqrt[3]{x-1}\), is a type of root function that finds the number which, when cubed, will give the value inside the function. Unique aspects of cube root functions include:

• Defined for all real numbers: Unlike square roots, which require the input to be non-negative, cube roots have no such restriction.
• They produce both positive and negative results: The cube root of a negative number is also negative, which distinguishes it from even root functions like square roots.
• Graph: The graph of a cube root function spans all four quadrants of a coordinate plane and passes through the origin.

In function composition, handling cube root functions requires us to understand their unrestricted domains. This understanding facilitates the seamless integration with other functions like linear functions, as seen in our exercise.

Linear Function

A linear function is one of the simplest types of functions, characterized by its straight-line graph. In our exercise, the final function \(f(g(x)) = x\) is a linear function, as it simplifies to \(y = x\). Key features of linear functions include:

• Constant rate of change: The slope of a linear function is constant, indicating a uniform rate at which \(y\) changes with respect to \(x\).
• No restrictions on domain: Linear functions are defined for all real values, \((-\infty, \infty)\), which means they are valid for every real number input.
• Graph: The graph of a linear function is a straight line, typically described by the formula \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

In the context of our composition function, recognizing that \(f(g(x))\) simplifies effectively to a linear function \(y = x\) allows us to confidently assert that the domain remains unrestricted. This corresponds to the integral concept that linear functions form the backbone of many complex mathematical models.