Problem 42 - Precalculus with Limits

Question

In Exercises 41-48, find (a) \(f \circ g\), and (b) \(g \circ f\). Find the domain of each function and each composite function. \(f(x) = \sqrt[3]{x-5}\), \(g(x) = x^3 + 1\)

Step-by-Step Solution

Answer

The composite function \(f \circ g\) is \(\sqrt[3]{x^3 - 4}\) with a domain of all real numbers, and \(g \circ f\) is \(x - 4\) with a domain of \([5,+\infty)\)

Step 1: Finding the composition \(f \circ g\)

We carry out the composition \(f(g(x))\), which means we substitute \(g(x)\) into the function \(f(x)\). i.e. \(f(g(x)) = f(x^3 + 1) = \sqrt[3]{(x^3 + 1) - 5} = \sqrt[3]{x^3 - 4}\)

Step 2: Identify the domain of \(f \circ g\)

The domain of the function \(f \circ g\) is the set of all values of \(x\) for which the composite function is defined. Any real number can be cubed and any number subtracted by 4 is also real. Therefore, the domain of \(f \circ g\) is all real numbers.

Step 3: Finding the composition \(g \circ f\)

We carry out the composition \(g(f(x))\), which means we substitute \(f(x)\) into the function \(g(x)\). \(g(f(x)) = g(\sqrt[3]{x-5}) = ((\sqrt[3]{x-5})^3) + 1 = (x-5) + 1 = x - 4\)

Step 4: Identify the domain of \(g \circ f\)

The cube root function is defined for all real numbers, so we need to find all \(x\) for which \(x-5\) is a real number. This means \(x\) must be greater or equal to 5. Therefore, the domain of \(g \circ f\) is \([5, +\infty)\)

Key Concepts

Domain of a Function

Understanding the domain of a function is essential, as it tells us all the possible input values for which the function is defined. The domain depends on the type of function we are dealing with. For example, with polynomial functions like quadratic, cubic etc., we typically consider all real numbers. However, for functions that involve square roots or divisions, we must ensure the expressions inside those roots or denominators are valid (non-negative or non-zero respectively).

Here are some key points to remember when determining the domain of different functions:

• For a function with a square root, ensure that the expression under the root is not negative.
• For functions involving fractions, the denominator should never be zero.
• Cubic roots and polynomial functions like cubic functions are generally defined for all real numbers, having no such restrictions.

Applying this to composite functions like in our exercise, the domain of the composite function is determined by both the domains of the composing functions and any additional restrictions that might come when combining them.

Function Composition

Function composition is an operation that takes two functions and produces another function. If you have two functions, say, \(f(x)\) and \(g(x)\), composing them involves creating a new function by applying one function to the results of the other. The notation \(f \circ g\) represents \(f(g(x))\), meaning you apply \(g\) first, then \(f\).

Here's how you can think about composing functions:

• Order of operations: Always apply the inner function first. In \(f \circ g\), start with \(g(x)\).
• Checking domains: Ensure that the output of the first function used as an input fits within the domain of the second function.
• Output creation: The result is another function that might have its unique domain based on how the initial domains interact.

By understanding function composition well, solving problems like those in the exercise on composing \(f(x) = \sqrt[3]{x-5}\) and \(g(x) = x^3 + 1\) becomes straightforward.

Cubic Functions

Cubic functions are polynomial functions of degree three, generally expressed in the form \(f(x) = ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants with \(a eq 0\). These functions have several unique characteristics that differentiate them from other polynomial degrees:

Some features of cubic functions include:

• End behavior: As \(x\) approaches infinity, the value of \(f(x)\) follows the sign of the leading coefficient \(a\) and vice versa as \(x\) approaches negative infinity.
• Graph shape: The graph of a cubic function can have up to two turning points and may not be symmetrical.
• Real and Complex roots: They always have at least one real root. However, all three roots may not be real.
• Domain and Range: The domain and range for a basic cubic function is all real numbers.

In the exercise, the cubic function \(g(x) = x^3 + 1\) is crucial for determining the composite with \(f(x)\). Its lack of restrictions on inputs helps create a broader domain for the composite \(f \circ g\).