Problem 24 - 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=\sqrt[3]{x}$$

Step-by-Step Solution

Answer

The inverse function \(f^{-1}(x)\) for the function \(f(x) = x^{1/3}\) is \(f^{-1}(x) = x^{3}\). Verification shows that both \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\), verifying that they are indeed inverse functions of each other.

Step 1: Finding the inverse function

To find the inverse of the function \(f(x)= x^{1/3}\), replace \(f(x)\) with \(y\). This gives you the equation \(y= x^{1/3}\). Swap \(x\) and \(y\) to get \(x = y^{1/3}\). Now, solve for \(y\). The equation becomes \(y = x^{3}\). Thus, \(f^{-1}(x) = x^{3}\).

Step 2: Verifying the correctness of the inverse function

Now, we need to verify that \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\). For \(f(f^{-1}(x))\), the function will look like \(f(x^{3})\), which simplifies to \((x^{3})^{1/3}\) and then to \(x\). For \(f^{-1}(f(x))\), the function will look like \((x^{1/3})^{3}\), which simplifies to \(x\). Therefore, the inverse function is correct and the function \(f(x)\) and its inverse are indeed true inverses.

Step 3: Brushing Up

This analysis illustrates the process of finding the inverse of a function and the importance of checking the results by substituting the function and its inverse into each other's equations. If – after this substitution – you get the same number you've inserted in the beginning, it confirms that you've found the correct inverse function.

Key Concepts

Inverse Function Verification

Understanding the concept of inverse function verification is crucial for grasping how functions can be reversed and applied in various mathematical contexts. When we talk about verifying an inverse function, we're essentially checking if the original function and its purported inverse are perfect mirrors of one another across the line y = x.

To perform this verification, you take the following essential steps: First, you re-evaluate the original function with the inverse function plugged into it—mathematically, this reads as f(f^{-1}(x)). If you've done everything right, the output will be simply x. Secondly, you feed the inverse function with the original function, notated as f^{-1}(f(x)). Again, if the inverse is correctly found, the result will be x. Verification with both approaches is a strong confirmation that the functions are indeed inverses.

Importantly, the procedure described ensures that the domain and range have been correctly swapped in the inverse, which is a foundational attribute of inverse functions. Without this thorough check, there's a risk of incorrectly assuming that two functions are inverses when they are not truly 'undoing' each other's operations.

One-to-One Functions

One-to-one functions, also known as injective functions, play an indispensable role in the concept of inverse functions. A function is one-to-one if each input corresponds to exactly one output and no output is the result of more than one input. What makes this property so significant?

Firstly, this uniqueness criterion enables the existence of an inverse because there won't be any ambiguity when reversing the function; each output in the range is linked to a particular input in the domain. For illustrative purposes, consider a simple function such as f(x) = 2x. For any given value of x, there is a distinct output, and no two different values of x will result in the same output value.

As a general test for whether a function is one-to-one, you can use the horizontal line test. If a horizontal line intersects the graph of the function at no more than one point, then it passes the test, confirming that the function is indeed one-to-one and that an inverse could be calculated for it. The presence of a function being one-to-one is what allows us to confidently proceed with finding its inverse, knowing that the inverse will also have this essential property.

Cubic Root Function

The cubic root function is a particular type of function in mathematics that undoes the action of a cubic function. If you have a function where f(x) = x^3, the cubic root function serves as its inverse and is notated as f^{-1}(x) = \( \sqrt[3]{x} \). Its role is to find the number, which when cubed, gives the original value. This function is an example of a broader class of root functions with critical applications in both pure and applied mathematics.

When dealing with cubic root functions, there are a few key properties to remember. They can handle any real number, positive or negative, as opposed to square root functions, which only deal with non-negative inputs. Furthermore, these functions are continuous and differentiable everywhere on the real number line, which means you can graph them without lifting your pencil from the paper.

Understanding the behavior of cubic root functions, their graphs, and their inverses is fundamental to many areas in mathematics and other STEM fields like physics and engineering. They help describe phenomena where there is a naturally cubic relationship between variables, and knowing how to utilize them expands the tools at your disposal for tackling complex problems.