Problem 84 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The value of \( f^{-1}(-8) \) is \(-2\).

Step 1: Understand the Inverse Function

The inverse function, denoted as \( f^{-1} \), essentially reverses the effect of the original function \( f(x) \). Here, we have \( f(x) = x^3 \). To find the inverse, we need a \( y \) value such that when input into \( f(x) \), it outputs the given number, in this case, \(-8\).

Step 2: Set Equation for Inverse

Since \( f(x) = x^3 \), the inverse function \( f^{-1}(y) \) will be such that \( x^3 = y \). For the specific inverse value \( f^{-1}(-8) \), set up the equation \( x^3 = -8 \).

Step 3: Solve for x

To find \( x \), solve the equation \( x^3 = -8 \). Take the cube root of both sides to find \( x = \sqrt[3]{-8} \).

Step 4: Simplify the Cube Root

The cube root of \(-8\) is a real number. We know \((-2)^3 = -8\), so \( \sqrt[3]{-8} = -2 \).

Step 5: Confirm Solution

Verify that inserting \( -2 \) back into the original function yields the correct output. Since \((-2)^3 = -8\), our solution satisfies the original equation.

Key Concepts

Cube Root

When dealing with equations where a number is raised to the third power, you might need to calculate the cube root. A cube root is the number that, when multiplied by itself three times, gives the original value. For instance, to find the cube root of \(-8\), you're looking for a number \(x\) which satisfies \(x^3 = -8\). Consider the following:

• The cube root of \(27\) is \(3\) because \(3 \times 3 \times 3 = 27\).
• Negative numbers also have real cube roots. For instance, the cube root of \(-8\) is \(-2\) as \((-2) \times (-2) \times (-2) = -8\).

In solving equations, recognizing how to correctly find and simplify the cube root is key, especially when dealing with inverse functions or any exponential equations.

Function Notation

Function notation is a way to represent a function in mathematics. It offers a clear and concise way of expressing the relationship between inputs and outputs. Let's break down the essentials:

• Function notation uses the format \(f(x)\), where \(f\) represents the function name and \(x\) is the variable or input.
• For example, in the function \(f(x) = x^3\), \(f(2)\) means you're substituting \(2\) into the function for \(x\), resulting in an output of \(8\) because \(2^3 = 8\).

A key part of decision-making in mathematics is understanding these notations. It allows one to manipulate function expressions confidently and assess the effects of their inverse, such as \(f^{-1}(y)\), which reverses the function's operations.

Solving Equations

Solving equations involves finding the value of the variable that makes the mathematical statement true. Different types of equations require various methods of solving. Let's see some steps in solving cube root equations:

• Start by recognizing the equation format. If you have \(x^3 = -8\), note that it calls for finding the cube root.


• Apply the cube root principle by taking the cube root on both sides of the equation to isolate \(x\) (i.e., \(x = \sqrt[3]{-8}\)). This simplification reveals \(x = -2\).

Solving equations can sometimes require checking the solution by substituting the value back into the original equation. This is crucial for verifying results, ensuring calculations are precise and accurate, and providing a real-world system check.