Problem 18 - College Algebra

Question

The functions in Exercises \(11-30\) 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 of the function \(x^3 - 1\) is \(\sqrt[3]{x + 1}\), and this has been verified.

Step 1: Find the inverse

To find the inverse of the function, let's denote \(f(x) = y\), so we have \(y = x^3 - 1\). Swap x and y to get \(x = y^3 - 1\). Solve for \(y\) to get the inverse function, \(f^{-1}(x)\). So, we get \(f^{-1}(x) = \sqrt[3]{x + 1}\)

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

Substitute \(f^{-1}(x)\) into the original function \(f(x)\). So, we substitute \(\sqrt[3]{x + 1}\) for \(x\) in \(f(x) = x^3 - 1\), and simplify. The simplified result is \(x\), so this equation is verified.

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

Substitute \(f(x)\) into \(f^{-1}(x)\). So, we substitute \(x^3 - 1\) for \(x\) in \(f^{-1}(x) = \sqrt[3]{x + 1}\), and simplify. The simplified result is \(x\), so this equation is verified also. Thus, \(\sqrt[3]{x + 1}\) is indeed the inverse of the given function.

Key Concepts

One-to-One Functions

A one-to-one function, also known as an injective function, is a special type of function where each element of the function's domain (input) is mapped to a unique element of the codomain (output). In simpler terms, no two different input values produce the same output. This distinct mapping ensures that the function has an inverse.
For example, consider the function \( f(x) = x^3 - 1 \). Each value of \( x \) in the domain gives a unique output, which means that no two different cubes, when reduced by 1, result in the same value. This characteristic confirms that \( f(x) \) is one-to-one.

• To determine if a function is one-to-one, you can use the horizontal line test: If any horizontal line intersects the graph of the function at most once, the function is one-to-one.
• If a function is one-to-one, it has an inverse that will allow you to work backwards from output to input.

Function Verification

Function verification involves checking whether a given inverse function truly represents the inverse of a specific function. This is an essential step to ensure that the work done in finding the inverse is correct.
For the original exercise, we need to verify that \( f^{-1}(x) = \sqrt[3]{x + 1} \) is indeed the inverse of the function \( f(x) = x^3 - 1 \). To do this, we use two key conditions:

• \( f(f^{-1}(x)) = x \): Substitute the inverse into the original function and simplify. This substitution should return the original input \( x \).
• \( f^{-1}(f(x)) = x \): Substitute the original function into the inverse and simplify. This should also return the original input \( x \).

If both conditions hold true, the inverse function is verified as correct, confirming its role in reversing the operation of the original function.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to solve equations or to transform them into a desired format.
When finding the inverse of a function, algebraic manipulation plays a crucial role. To find the inverse of \( f(x) = x^3 - 1 \), we start by setting \( y = x^3 - 1 \) and then solve for \( x \):
1. Swap \( x \) and \( y \), giving us \( x = y^3 - 1 \).
2. Solve for \( y \) by adding 1 to both sides and taking the cube root: \( y = \sqrt[3]{x + 1} \).
3. Thus, the inverse function is \( f^{-1}(x) = \sqrt[3]{x + 1} \).

• Algebraic manipulation helps in isolating the variable of interest (here, \( y \)) so that we can clearly express the inverse.
• Understanding algebraic manipulation involves following basic arithmetic rules while strategically rearranging the expressions.

Mastering this skill simplifies complex problems and aids in comprehending function inverses more effectively.