Problem 60 - Algebra and Trigonometry

Question

Find the inverse of \(f(x)=x^{3}+2.\)

Step-by-Step Solution

Answer

The inverse of the function \(f(x)=x^{3}+2\) is \(f^{-1}(x) = \sqrt[3]{x - 2}\).

Step 1: Identify and replace the function

The function given is \(f(x) = x^{3} + 2\). To find its inverse, you replace the function notation \(f(x)\) with the variable \(y\). This gives us \(y = x^{3} + 2\).

Step 2: Swap \(x\) and \(y\)

The next step is interchanging \(x\) and \(y\). Therefore, the equation becomes \(x = y^{3} + 2\). This helps us rewrite the function in terms of \(y\).

Step 3: Solve for \(y\)

This step involves isolating \(y\) to one side. Therefore, subtract 2 from both sides of the equation. This provides \(x - 2 = y^{3}\). To isolate \(y\), take the cube root of both sides. This results in the inverse function \(y = \sqrt[3]{x - 2}\).

Step 4: Write the inverse function

Replace \(y\) with \(f^{-1}(x)\) (the notation denoting the inverse of the function). Final answer is \(f^{-1}(x) = \sqrt[3]{x - 2}\).

Key Concepts

Cube Root

When we talk about cube roots, we're dealing with the idea of finding which number, when multiplied by itself three times, produces the given number. It is the opposite operation of cubing a number.
For example, if you have the expression \(3\), raising \(3\) to the power of three gives us \(27\), and the cube root of \(27\) is \(3\). In mathematical terms, we denote the cube root using the radical sign with a small three: \(\sqrt[3]{x}\).

• The cube root of a number \(a\) is denoted as \(b = \sqrt[3]{a}\), meaning \(b^3 = a\).
• Cube root can be applied to solve problems involving cubes, where you need to "undo" a cubing operation.
• This concept is especially useful while dealing with inverse functions, as it reverses the action of cubing.
• Cube roots are defined for all real numbers since multiplying a real number by itself three times always results in a real number.

Understanding cube roots can help simplify complex equations, and it's key to finding inverse functions of cubic equations.

Function Notation

Function notation is a way to express the relationship between variables, often in the form of \(f(x)\). This notation helps to define a function in specific terms and is very useful in mathematics.
In our original exercise, we're given \(f(x) = x^{3} + 2\), which represents a function named \(f\) with \(x\) as the independent variable.

• Function notation is read as "\(f\) of \(x\)," where \(f(x)\) represents the value of the function at \(x\).
• Using function notation helps identify how an input value \(x\) is converted to an output value \(f(x)\).
• We can also use it to denote an inverse function: replacing \(f(x)\) with \(f^{-1}(x)\) shows the reverse operation of the original function.
• This notation simplifies complex expressions and improves the clarity of functions in mathematics.

In the solution provided, we saw how replacing \(f(x)\) with \(y\) assisted in finding the inverse. Once found, we switch back to \(f^{-1}(x)\), keeping track of what the function represents.

Solving Equations

Solving equations involves finding the value(s) of variables that make the equation true. It is a fundamental skill in mathematics and is key to understanding many mathematical concepts, especially inverses.
The original exercise demonstrates solving for \(y\) by manipulating the equation \(x = y^{3} + 2\) to isolate \(y\). This process includes:

• Isolating \(y\) by performing inverse operations, such as subtracting \(2\) from both sides, resulting in \(x - 2 = y^{3}\).
• Applying the cube root to both sides to "undo" cubing \(y\), yielding \(y = \sqrt[3]{x - 2}\).

These steps illustrate the methodical approach needed when working through equations, which often requires doing the opposite of the operations initially applied to solve for the variable.
Solving equations like these helps clarify the process and ensures that the relationships between variables are maintained, enabling us to derive inverse functions accurately. Understanding these steps equips you with the tools to approach more complex mathematical problems.