Problem 74 - A Graphical Approach to College Algebra

Question

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=4-3 x^{3}$$

Step-by-Step Solution

Answer

The inverse is \(f^{-1}(x) = \sqrt[3]{\frac{4 - x}{3}}\).

Step 1: Understand One-to-One Functions

A function is considered one-to-one (or injective) if different inputs give different outputs. This means for every output there is a unique input. Thus, the function should be able to have an inverse.

Step 2: Write the Equation for Finding the Inverse

To find the inverse function, first set the equation for the original function: \(y = 4 - 3x^3\). To find the inverse, solve this equation for \(x\).

Step 3: Solve for \(x\) in Terms of \(y\)

Rearrange the equation to isolate \(x\):1. Move 4 to the other side: \(-3x^3 = y - 4\).2. Divide both sides by -3: \(x^3 = \frac{4 - y}{3}\).3. Take the cube root of both sides: \(x = \sqrt[3]{\frac{4 - y}{3}}\).

Step 4: Express the Inverse Function

Having solved for \(x\) in terms of \(y\), the inverse function is obtained by replacing \(y\) with \(x\). Thus, \(f^{-1}(x) = \sqrt[3]{\frac{4 - x}{3}}\).

Key Concepts

One-to-One Functions

One-to-One functions, also known as injective functions, are special types of functions where each input corresponds to a unique output. This means no two different inputs will produce the same output.

• For example, if you have a function where $f(a) = f(b)$, then it must be that $a = b$ for the function to be one-to-one.
• This special property allows one-to-one functions to have inverse functions, as each output corresponds to exactly one input, making the mapping reversible.

Knowing whether a function is one-to-one is crucial when solving problems involving inverses. To verify if a function is one-to-one, the horizontal line test can be applied. If a horizontal line crosses the graph of the function more than once, then the function is not one-to-one.

Cube Root

The cube root is an operation that finds a number which, when multiplied by itself twice, results in the original number. Mathematically, the cube root of a number \(a\) is denoted as \(\sqrt[3]{a}\).
The cube root function has some interesting properties:

• It can accept negative numbers. For example, the cube root of \(-8\) is \(-2\) because \((-2) \times (-2) \times (-2) = -8\).
• This property makes it unique compared to the square root function, which only applies to non-negative numbers unless we venture into complex numbers.

In solving inverse functions involving cube roots, remember that the cube root operation is essential in reversing the process of cubing a number. For instance, when rearranging a function to solve for \(x\) in terms of \(y\), taking the cube root can help isolate the variable. This is a fundamental step, as seen in our solution.

Function Notation

Function notation is a way to represent functions clearly and unambiguously. Functions are often written in the form \(f(x)\), where \(f\) is the name of the function, and \(x\) is the variable.
This notation helps in:

• Indicating the output: For \(f(x)=4-3x^3\), \(f(x)\) represents the output when \(x\) is the input to the function \(f\).
• Working with inverse functions: The notation \(f^{-1}(x)\) is used to represent the inverse of the function \(f(x)\).

Using function notation allows mathematicians and students alike to easily navigate and manage complex expressions, ensuring clarity. It also helps when switching roles of variables: to express an inverse like \(f^{-1}(x)=\sqrt[3]{\frac{4-x}{3}}\), we replace \(y\) with \(x\) after finding the expression of inverse.