Problem 18 - College Algebra Essentials

Question

The functions in Exercises \(11-28\) 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 \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\) $$f(x)=x^{3}-1$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = (x+1)^{1/3}\). This has been verified by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\).

Step 1: Finding the Inverse Function

The given function is \(f(x)=x^{3}-1\). To find its inverse \(f^{-1}(x)\), we need to interchange x and y and solve for y. For our function, \$f(x) = y = x^3 - 1\$, we interchange x and y to obtain \$x = y^3 - 1\$. Solving this for y gives \(f^{-1}(x) = (x+1)^(1/3)\).

Step 2: Verifying the Inverse Function

To verify if this is the correct inverse function, we need to show that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). Substituting \(f^{-1}(x)\) into the function gives \(f\left(f^{-1}(x)\right) = ((x + 1)^\frac{1}{3})^{3} - 1 = x\). And substituting \(f(x)\) into the inverse gives \(f^{-1}(f(x)) = (x^{3}-1+1)^\frac{1}{3} = x\). Since both evaluations equal x, the inverse is correct.

Key Concepts

One-to-One Functions

Understanding one-to-one functions is crucial for delving into the world of inverse functions. A one-to-one function, also known as an injective function, is a type of function where each element of the range is paired with exactly one unique element of the domain.

In formal terms, a function \( f \) is considered one-to-one if, given any two different elements \( a \) and \( b \) from the domain, \( f(a) \) does not equal \( f(b) \). This means no horizontal line can intersect the graph of the function more than once.

For example, the function given by \( f(x)=x^{3}-1 \) is one-to-one because for each output, there is a distinct input value. Visualizing this graphically or testing a few values can confirm its one-to-one nature. This property is fundamental when we want to find an inverse because only one-to-one functions have inverses that are also functions.

Verifying Inverse Functions

Verifying inverse functions involves a couple of simple, yet powerful checks. After finding a suspected inverse, we need to confirm it genuinely reverses the action of the original function.

An inverse function, denoted as \( f^{-1}(x) \), essentially undoes whatever the original function \( f(x) \) does. To verify that you've correctly found an inverse, you must show two things:

• \( f(f^{-1}(x)) = x \) for every x in the range of \( f^{-1} \)
• \( f^{-1}(f(x)) = x \) for every x in the domain of \( f \)

This is akin to saying if you follow a path to a certain point and then retrace your steps back, you should end up at your starting point. Applying these checks to our previous example function, the calculations showed that both conditions held true, confirming the correctness of the inverse function found.

Equation Solving in Inverse Functions

The process of equation solving in the context of finding an inverse function involves a few transformational steps.

Initially, you start with the function \( f(x) \), which in our case, is \( f(x) = x^{3} - 1 \). The goal is to express \( x \) in terms of \( y \) (or the other way around, depending on the perspective). This requires us to switch the roles of \( x \) and \( y \), because we're looking at the function from its output back to its input.

Once you've interchanged the variables, the next step is to solve the resulting equation for the new 'output' variable. In the example, swapping and solving gave us \( f^{-1}(x) = (x+1)^{\frac{1}{3}} \). This approach is standard for finding inverses and proves to be a systematic method for tackling such problems. It's important, though, to remember that not all functions have inverses—only one-to-one functions do, and the process detailed here is applicable solely to them.