Problem 24 - Precalculus Essentials

Question

The functions are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=\sqrt[3]{x}$$

Step-by-Step Solution

Answer

The inverse of the function \(f(x)=\sqrt[3]{x}\) is \(f^{-1}(x)=x^3\). The verification parts confirm that it's the correct inverse as \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\), both simplified to \(x\).

Step 1: Finding the Inverse

Express \(f(x)=\sqrt[3]{x}\) as \(y=\sqrt[3]{x}\) and then swap \(x\) and \(y\) to get \(x=\sqrt[3]{y}\). Then, cube both sides to solve for \(y\), which gives us \(y=x^3\). Thus, \(f^{-1}(x)=x^3\).

Step 2: Verification of the Inverse Pair - Part A

For \(f(f^{-1}(x)) = x\), substitute \(f^{-1}(x)\) with the obtained function \(x^3\), into the function \(f(x)\). This gives us \(f(f^{-1}(x)) = (\sqrt[3]{x})^3 = x\), which verifies the first part of the validation.

Step 3: Verification of the Inverse Pair - Part B

For \(f^{-1}(f(x)) = x\), substitute \(f(x)\) with \(\sqrt[3]{x}\), into the function \(f^{-1}(x)\). This gives us \(f^{-1}(f(x)) = (\sqrt[3]{x})^3 = x\), which verifies the second part of validation.

Key Concepts

One-to-One Functions

A function is described as "one-to-one" if it matches each element of its domain to a distinct element in its range. This means that no two different inputs share the same output. Such types of functions are important because only one-to-one functions have inverses that are also functions.

Some key properties of one-to-one functions include:

• Each value in the range corresponds to exactly one value in the domain.
• A horizontal line intersects the graph of a one-to-one function at most once.
• If we have two different values, say \(a\) and \(b\), then \(f(a) eq f(b)\).

To determine if a function is one-to-one, we can use the Horizontal Line Test. If a horizontal line cuts the graph in more than one place, it means the function is not one-to-one. In the exercise above, the cube root function \(f(x)=\sqrt[3]{x}\) is one-to-one because each input of \(x\) gives a unique output that is not repeated.

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. In simpler words, if you have two functions, say \(f(x)\) and \(g(x)\), the composition \((f \circ g)(x)\) means \(f(g(x))\). Function composition is a powerful tool because it allows us to perform multiple operations at once.

In terms of inverse functions, composition can also check if two functions are inverses by using the identities:

• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

The idea is that applying one function and then immediately applying its inverse will return the original input. In the exercise, we verified that for \(f(x)=\sqrt[3]{x}\) and its inverse \(f^{-1}(x)=x^3\), both of these identities hold true, confirming that they are indeed inverses of each other.

Cube Root Function

The cube root function is a specific type of root function expressed as \(f(x) = \sqrt[3]{x}\). It has unique properties that make it interesting and useful in both algebra and calculus.

Some characteristics of the cube root function are:

• Unlike the square root function, the cube root is defined for all real numbers, including negatives.
• The graph of \(\sqrt[3]{x}\) is a curve that passes through the origin (0,0) and extends indefinitely in both directions.
• It is a one-to-one function, so it has an inverse: \(f^{-1}(x) = x^3\).

In the exercise, using the cube root function, we found its inverse by reversing the process: if \(y = \sqrt[3]{x}\), swapping \(x\) and \(y\) gives \(x = \sqrt[3]{y}\), and cubing both sides gives \(y = x^3\). This step-by-step manipulation highlights not only the procedure but also why cube root functions have straightforward and accessible inverses.