Problem 21 - Precalculus with Limits

Question

In Exercises 19-22, verify that \(f\) and \(g\) are inverse functions. \(f(x) = x^3 + 5\), \(g(x) = \sqrt[3]{x-5}\)

Step-by-Step Solution

Answer

Based on the computation in step 1 and step 2, both conditions are verified and thus, \(f\) and \(g\) are indeed inverse functions.

Step 1: Verify the condition \(f(g(x)) = x\)

Substitute \(x\) for \(g(x)\) in \(f(x)\): \(f(g(x)) = (g(x))^3 + 5 = (\sqrt[3]{x-5})^3 + 5 = x - 5 + 5 = x.\) Therefore, the first condition \(f(g(x)) = x\) is verified.

Step 2: Verify the condition \(g(f(x)) = x\)

Substitute \(x\) for \(f(x)\) in \(g(x)\): \(g(f(x)) = \sqrt[3]{f(x) - 5} = \sqrt[3]{(x^3 + 5) - 5} = \sqrt[3]{x^3}= x.\) Therefore, the second condition \(g(f(x)) = x\) is also verified.

Key Concepts

Composite Functions

Composite functions are functions made by combining two or more functions. In our exercise, we are using two functions, \(f(x)\) and \(g(x)\), and are checking if they are inverses by verifying the composition \(f(g(x)) = x\) and \(g(f(x)) = x\). This process ensures that the output of one function becomes the input of the other, returning the original input value. In other words, when a function is applied and then its inverse is applied, you end up where you started. This ability to invert functions using their composition is crucial for confirming that two functions truly are inverses of each other, as shown in the step-by-step solution.

Cubic Functions

Cubic functions are polynomial functions where the highest degree of the argument is three, symbolized as \(x^3\). These functions can take on various forms, but in general, from an algebraic perspective, they are exactly represented by \(f(x) = x^3 + c\), where \(c\) is a constant. In our exercise, the cubic function is \(f(x) = x^3 + 5\). Cubic functions are essential in providing flexibility within function modeling as they can describe more complex, non-linear relationships. Since they have an 'odd' order of power, cubic functions have properties like always having at least one real root and behaving similarly in both positive and negative directions, although in different quadrants.

Cube Root Functions

Cube root functions inverse cubic functions. The general form of a cube root function is \(g(x) = \sqrt[3]{x - c}\), where \(c\) is a constant. For the function \(g(x) = \sqrt[3]{x-5}\) from our exercise, it specifically undoes the action of the given cubic function \(f(x) = x^3 + 5\).

• Cube root functions map each real number directly to one real output, unlike square root functions which can handle only non-negative numbers.
• They retain the domain and range symmetry about the origin.
• In essence, cube root functions are straightforward and unambiguous mathematical inverses of cubic functions, ensuring that the output of a cube function becomes the input for its root transformation and vice versa.

Cube root functions provide an algebraic mirror of the behavior of cubic functions as they allow us to retrieve an original input value through the process of inversion.