Problem 18 - 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=x^{3}-1$$

Step-by-Step Solution

Answer

The inverse function, \(f^{-1}(x)\), of the given function \(f(x) = x^{3} - 1\) is \(f^{-1}(x) = (x+1)^{1/3}\). The correctness of \(f^{-1}(x)\) has been established by verifying both \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Step 1: Find the Inverse Function

The given function is \(f(x) = x^{3} - 1\). To find its inverse, we interchange \(x\) with \(y\) to get \(x = y^{3} - 1\), and then solve for \(y\). Doing so, we get \(y = (x+1)^{1/3}\). This gives us the inverse function \(f^{-1}(x) = (x+1)^{1/3}\).

Step 2: Verify \(f(f^{-1}(x)) = x\)

We need to substitute \(f^{-1}(x)\) into \(f(x)\), i.e., \(f((x+1)^{1/3})\). As per the definition of \(f(x)\), we substitute \(x = (x+1)^{1/3}\) in \(f(x) = x^{3} - 1\), we get \( ((x+1)^{1/3})^{3} - 1 = x + 1 - 1 = x\). Therefore, \(f(f^{-1}(x)) = x\), confirming the correctness of the derived inverse function.

Step 3: Verify \(f^{-1}(f(x)) = x\)

In this step, we substitute \(f(x)\) into \(f^{-1}(x)\), i.e., \(f^{-1}(x^3 - 1)\). As per the definition of \(f^{-1}(x)\), we substitute \(x = x^{3} - 1\) in \(f^{-1}(x) = (x+1)^{1/3}\), which gives us \((x^{3} - 1 + 1)^{1/3} = x^{1/3*3} = x\). Therefore, \(f^{-1}(f(x)) = x\), which again validates the derived inverse function.

Key Concepts

One-to-one functions

A one-to-one function, sometimes called an injective function, is a function where each input corresponds to a unique output. This means no two different input values will produce the same output. Understanding this characteristic is crucial because only one-to-one functions have well-defined inverses.

To check if a function is one-to-one, you can use the Horizontal Line Test. Simply graph the function and see if any horizontal line intersects the graph more than once. If not, it's one-to-one. Also, an algebraic method involves assuming that if two outputs are equal, then their corresponding inputs must also be equal. If this can be proven, the function is one-to-one.

Here are some traits of one-to-one functions:

• They pass the Horizontal Line Test.
• Each unique input grants a unique output.
• They have inverses because they are bijections.

Recognizing one-to-one functions sets the foundation for finding inverse functions. In our problem, the function given, which is a cubic function, is one-to-one, allowing us to find its inverse.

Function verification

Once an inverse function is calculated, it's important to verify its correctness. Verification ensures that the inverse you found is mathematically sound and true.

Verification relies on two essential properties that should hold:

• \( f(f^{-1}(x)) = x \): This confirms substituting the inverse function into the original function returns the original input, \(x\).
• \( f^{-1}(f(x)) = x \): This verifies applying the original function and then the inverse returns to the original input, \(x\).

In the original solution, this was demonstrated by substituting the inverse back into the function and vice versa. Doing both checks confirms the validity of an inverse.

Verification is vital in mathematics as it confirms accuracy. In this exercise, both conditions held true, affirming the inverse derived \(f^{-1}(x) = (x+1)^{1/3}\) is correct.

Cubic functions

Cubic functions have the general form \(f(x) = ax^3 + bx^2 + cx + d\). These functions are named for their degree of three, represented by the highest power of \(x\), which is cubed. The function in the exercise, \(f(x) = x^3 - 1\), is an example of a basic cubic function without extra linear or quadratic terms.

Cubic functions are intriguing because of their distinct graphs that can have up to two turning points and can either increase or decrease to infinity in both directions. Due to their complexity, they often have distinct behaviors such as symmetry and inflection points. Fortunately, many cubic functions, like our given function \(f(x) = x^3 - 1\), are inherently one-to-one making it possible to find inverses.

Understanding the characteristics of cubic functions allows one to explore their one-to-one nature easily. These include:

• Graph shapes vary widely, but many are one-to-one.
• Symmetries or pivots around certain points.
• Finding inverses involves algebraic manipulation, often requiring solving equations like \(y = x^3 - 1\) and rearranging to \(x = y^{3/1} + 1\).

Being able to recognize and work with cubic functions opens up a pathway to understanding more complex algebraic structures.