Problem 17 - 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=x^{3}+2$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x^3 + 2\) is \(f^{-1}(x) = \sqrt[3]{x - 2}\), and it has been verified through the equations \(f(f^{-1}(x)) = x \) and \(f^{-1}(f(x)) = x\).

Step 1: Finding the Inverse Function

To find the inverse of a function, one of the simplest ways is to replace \(f(x)\) with \(y\), and then swap \(x\) and \(y\). So, the given function \(f(x) = x^3 + 2\) becomes \(y = x^3 + 2\). Swapping \(x\) and \(y\) gives \(x = y^3 + 2\). Solving for \(y\) which gives us the inverse function \(f^{-1}(x)\), we subtract 2 from both sides to get \(x - 2 = y^3\), then take the cube root of both sides to get \(y = \sqrt[3]{x - 2}\). So the inverse function \(f^{-1}(x) = \sqrt[3]{x - 2}\).

Step 2: Verifying the Inverse Function Part 1

To verify the inverse, we need to prove that \(f(f^{-1}(x)) = x \). Substituting \(f^{-1}(x)\) into the original function \(f(x)\) gives \(f(f^{-1}(x)) = (\sqrt[3]{x - 2})^3 + 2\). Simplifying this equation gives \(x - 2 + 2\), which equals to \(x\). Thus, \(f(f^{-1}(x)) = x \) has been verified.

Step 3: Verifying the Inverse Function Part 2

The second part of the verification is showing that \(f^{-1}(f(x)) = x\). Substituting \(f(x)\) into the inverse function \(f^{-1}(x)\) gives \(f^{-1}(f(x)) = \sqrt[3]{(x^3 + 2) - 2}\). Simplifying this equation gives \(\sqrt[3]{x^3} = x\). Thus, \(f^{-1}(f(x)) = x\) has also been verified.

Key Concepts

Inverse Functions

Understanding inverse functions is akin to learning a new language backwards. It's about finding a function that essentially reverses another function's actions. Think of putting on and then taking off a coat; the processes are opposites or 'inverses' of each other.

Mathematically, to find an inverse function, we often start with a function like our example, f(x) = x^3 + 2. By replacing f(x) with y, as shown in the solution, we can more clearly see the relationship where y depends on x. When we swap x and y, we are setting up an equation where x now depends on y. Solving for y gives us the inverse f^{-1}(x) = \( \sqrt[3]{x - 2} \), which will reverse the operations of the original function.

To really understand the inverse, picture a box where you put in a number, the function does its magic, and you get a result. The inverse is another box where the result becomes input, and you get back the original number. Finding the inverse means creating the fancy machinery inside this second box.

Function Verification

Once we've crafted our inverse function, we need to ensure it's not a dud. That's where function verification plays a critical role. The idea is simple: if we've concocted the correct inverse function, it should undo whatever the original function did. Much like a lock and key, if the inverse fits, it verifies the function.

In our exercise, verification is done in two steps. First, we show that if we start with x, apply our inverse, and then the original function, we must end up back at x. This was illustrated by f(f^{-1}(x)) = x. The second step is similar but in reverse order: apply the original function to x, and then the inverse, which should also circle back to x, shown as f^{-1}(f(x)) = x. If both of these checks come out clean, we've verified that our inverse function isn't an imposter.

An essential tip for students is to perform these verification steps carefully. It ensures that the inverse truly undoes the action of the function and that no mathematical hocus-pocus has occurred along the way.

One-to-One Functions

Now, let's talk about one-to-one functions. This is a crucial concept because not every function is entitled to an inverse. For a function to have an inverse, it must be one-to-one, meaning each input is connected to one unique output, and vice versa.

Imagine a function as a party host handing out unique gift bags to each guest (input). If every guest gets a different bag, it's a one-to-one function. But if even two guests end up with the same bag, the function fails at being one-to-one. Visually, one-to-one functions pass the 'Horizontal Line Test': if any horizontal line crosses the function's graph more than once, the function isn't one-to-one.

Thankfully, in the given exercise, f(x) = x^3 + 2 is one-to-one, since every x gives a different f(x), and we can reverse this process uniquely. It ensures that finding an inverse is not only possible but also meaningful—guaranteeing that each output from the function can be traced back to one specific input.