Problem 29 - 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=\sqrt[3]{x-4}+3$$

Step-by-Step Solution

Answer

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

Step 1: Express the function

Consider the one-to-one function \(f(x)=\sqrt[3]{x-4}+3\).

Step 2: Find the inverse function

To find the inverse of this function, switch the roles of \(y\) and \(x\). Thus, the function becomes \(x=\sqrt[3]{y-4}+3\). Now, we need to solve this equation for \(y\). Starting by subtracting 3 from both sides, we get \(x-3=\sqrt[3]{y-4}\). Cubing both sides (to eliminate the cube root), we get \((x-3)^3 = y - 4\). Solving finally for \(y\), we get \(y = (x-3)^3 + 4\). This equation represents \(f^{-1}(x)\).

Step 3: Verify the equation

In order to verify this equation, you must show that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). 1) Substituting \(f^{-1}(x)\) into \(f(x)\), we get \(f(f^{-1}(x)) = \sqrt[3]{(x-3)^3+4-4}+3 = \sqrt[3]{(x-3)^3}+3 = x-3+3 = x\). 2) Substituting \(f(x)\) into \(f^{-1}(x)\), we get \(f^{-1}(f(x)) = ((\sqrt[3]{x-4}+3)-3)^3+4 = (\sqrt[3]{x-4})^3+4 = x-4+4 = x\). Thus, both verification steps hold true.

Key Concepts

One-to-One Functions

A one-to-one function, also known as an injective function, is a function where each output value is paired with exactly one input value. This means that different input values correspond to different output values. One important property of one-to-one functions is that they have an inverse that is also a function. In other words, you can "reverse" the function and find the original input given the output.

To determine if a function is one-to-one, you can use the Horizontal Line Test. If no horizontal line intersects the graph of the function more than once, then the function is one-to-one. One-to-one functions are essential because they allow us to find inverses, providing us the ability to reverse the process the function represents. This is crucial in many real-world applications, such as encryption and decryption in computer science, or converting measurements from one unit to another.

Verification of Inverse Functions

Verification of inverse functions ensures that the function you've determined as the inverse actually reverses the original function. This involves showing that when you compose the function and its inverse, you get the identity function, which is simply the input back again.

To verify an inverse, you need to check two compositions:

• First, verify that the original function applied to its inverse, also known as \(f(f^{-1}(x))\), results in \(x\).
• Second, check that the inverse function applied to the original function, given by \(f^{-1}(f(x))\), also gives back \(x\).

Both conditions must be satisfied for the inverse function to be correct. Following these steps properly confirms that your inverse operation is accurate, ensuring the function you've derived as the inverse accurately "undoes" the original function's operations.

Finding Inverses

Finding the inverse of a function involves swapping the dependent and independent variables and then solving for the new dependent variable. This transforms the function, allowing you to retrace each step it performs on an input.To find an inverse:

• Start by replacing \(f(x)\) with \(y\).
• Switch the roles of \(x\) and \(y\), rewriting the function as \(x\) as a function of \(y\).
• Solve this equation for \(y\) in terms of \(x\).

This resulting equation represents the inverse function \(f^{-1}(x)\). It's important to note that not all functions have inverses that are functions. Only one-to-one functions do. When successful, the inverse function essentially "unwinds" the original function, providing insights into the original transformation applied by \(f\).

Cube Root Functions

Cube root functions apply a cube root operation to the variable. The general form is \(f(x) = \sqrt[3]{x} + c\), where \(c\) is a constant. Cube root functions are particularly interesting because they have a domain of all real numbers and a range of all real numbers, meaning they can take any real number as an input and produce any real number as an output.To invert a cube root function, for example, \(f(x) = \sqrt[3]{x-4} + 3\), you have to reverse each step:

• First, subtract any added constants inside the function before taking the cube root.
• Next, undo the cube root by cubing both sides of the equation.
• Finally, solve for the independent variable.

This ensures that you precisely reverse the function's operations. Cube root functions, due to their simplicity and continuous nature, often serve as a basis for understanding more complex functions, enhancing comprehension of mathematical transformation processes.