Problem 82 - A Graphical Approach to College Algebra

Question

Let \(f(x)=x^{3} .\) Evaluate each expression. $$f^{-1}(8)$$

Step-by-Step Solution

Answer

\( f^{-1}(8) = 2 \) because \( 2^3 = 8 \).

Step 1: Understanding the Problem

We are given the function \( f(x) = x^3 \) and need to evaluate \( f^{-1}(8) \). This means we have to find the value of \( x \) such that \( f(x) = 8 \).

Step 2: Express the Inverse Function Relation

Since \( f(x) = x^3 \), the inverse function \( f^{-1}(y) \), if \( y = x^3 \), is given by solving \( x = y^{1/3} \). It reverses the operation of \( f(x) \).

Step 3: Substitute and Solve for the Inverse

To find \( f^{-1}(8) \), substitute \( y = 8 \) in \( x = y^{1/3} \). Thus, \( x = 8^{1/3} \).

Step 4: Compute the Cube Root

Calculate \( 8^{1/3} \). The cube root of 8 is the number which, when cubed, equals 8. This number is 2 because \( 2^3 = 8 \).

Step 5: Conclude the Solution

Thus, \( f^{-1}(8) \) is 2.

Key Concepts

Cube Root

The cube root of a number is an essential concept in mathematics. It represents a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because raising 2 to the power of 3 (or cubing it) equals 8: \(2^3 = 8\). Calculating cube roots is often necessary when working with inverse functions, particularly when a function involves a cubic expression, such as \(f(x) = x^3\). To understand cube roots more deeply, let's consider a few points:

• The cube root of a number \(a\) is often written as \(a^{1/3}\) or \(\sqrt[3]{a}\).
• Unlike square roots, cube roots can be evaluated for negative numbers. For instance, \((-8)^{1/3} = -2\) because \((-2)^3 = -8\).
• Cube roots help in reversing cubic operations, which is very useful for solving equations involving cubes.

By mastering the concept of cube roots, solving problems where you need to find the inverse of cubed functions becomes much easier.

Function Notation

Function notation is a systematic way of representing functions in mathematics. It uses symbols to denote the relationship between inputs and outputs. In our exercise, we explore the function \(f(x) = x^3\). This notation helps us to specify a particular process or rule that transforms an input \(x\) into an output. Let's delve into some key points about function notation:

• \(f(x)\) means that there is a function \(f\) where \(x\) is the variable input.
• An inverse function, such as \(f^{-1}(y)\), undoes the action of the original function. It maps outputs back to their original inputs.
• To find \(f^{-1}(8)\), we focus on finding the value of \(x\) that satisfies the equation when \(x\) is inputted into the inverse function.

This notation not only provides clarity but also serves as a guide for solving various types of equations, especially when interacting with the concept of inverses.

Solving Equations

When we solve equations, we aim to find the unknown variable that satisfies the given mathematical statements. In the context of our exercise, we need to solve for \(x\) when given an inverse function. The process usually follows a logical sequence of steps, breaking down any operations applied to the variable. Consider a few essential steps involved in solving equations:

• Identify the equation you need to solve, like \(f(x) = y\). In our example, we had \(x^3 = 8\).
• Apply algebraic operations to isolate the variable. For instance, to solve for \(x\), we apply cube root, resulting in \(x = 8^{1/3}\).
• Calculate and verify the solution. Here, \(8^{1/3} = 2\), verifying that \(2^3 = 8\).

Practicing these steps systematically can make solving even complex equations straightforward. This approach assures that you arrive at the correct solution, reinforcing your understanding of mathematical concepts.