Problem 65 - Algebra 2

Question

Find the inverse of each function. Is the inverse a function? $$ f(x)=\frac{\sqrt[3]{x}}{3} $$

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = \sqrt[3]{x}/3\) is \(f^-1(x) = (3x)^3\). This inverse is a function.

Step 1: Find the inverse

To find the inverse of a function \(f(x)\), we must replace \(f(x)\) with \(y\), then swap \(x\) and \(y\), and finally solve for \(y\).\n\nSo we start with \(y = \sqrt[3]{x}/3\) and swap \(x\) and \(y\) to get \(x = \sqrt[3]{y}/3\). Now we solve for \(y\). We start by multiplying by 3 to isolate the cube root. This gives us \(3x = \sqrt[3]{y}\). Next we cube both sides to get rid of the cube root on the right. This leaves us with \((3x)^3 = y\).

Step 2: Standard form of the inverse

In the previous step, we found the inverse function to be \((3x)^3 = y\). Now we put this in standard form as \(y = (3x)^3\). Therefore, the inverse function of \(f(x) = \sqrt[3]{x}/3\) is \(f^-1(x) = (3x)^3\).

Step 3: Check if the inverse is a function

To check if the inverse is a function, we must verify the vertical line test. This test states that if any vertical line intersects the graph of the relation at exactly one point, then the relation is a function. The equation \((3x)^3\) is a cubic function, which is known to satisfy the vertical line test. Thus, the inverse function is indeed a function.

Key Concepts

Cubic functions

Cubic functions are polynomial functions of degree three. The general form of a cubic function is given by \( f(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants and \( a eq 0 \). These functions are characterized by their distinctive curve shape which can span all four quadrants of the Cartesian plane.

A key feature of cubic functions is that they have one to three real roots, depending on the discriminant and the specific values of the coefficients. The graph shape is highly dependent on these coefficients, which influence the turning points and inflections of the curve.

When solving for the inverse of a cubic function, as shown in the original exercise, it's important to ensure each step is calculated properly. For example, the original function \( f(x) = \frac{\sqrt[3]{x}}{3} \) becomes \( f^{-1}(x) = (3x)^3 \) when finding the inverse. This demonstrates how a simple transformation can lead to a different polynomial relationship.

Cubic functions are particularly interesting because they always pass the vertical line test, confirming that they are valid functions. This inherent property makes their inverses equally significant, as seen in the function \( (3x)^3 \).

Vertical line test

The vertical line test is a quick and effective method for determining whether a graph represents a function. It involves drawing vertical lines across the graph. If at any point a line crosses the graph more than once, the graph does not represent a function.

For instance, when we apply this test to the inverse function \( (3x)^3 \) from our exercise, each vertical line crosses the graph only once. This confirms that the inverse function is valid. This happens because cubic functions, like the original function \( \frac{\sqrt[3]{x}}{3} \), naturally comply with the one-to-one requirement needed for functions.

The ability to use the vertical line test means you can quickly confirm the nature of your equations without complex calculations. It's an intuitive way to visualize and verify the functionality of a given equation efficiently.

Function notation

Function notation is a fancy way to express the relationship between variables. It's like giving a name to the process that takes an input \( x \) and gives back an output \( y \). When you see \( f(x) \), it's saying: "for this function, put this \( x \) value in, and get the result".

In our example, we started with the function \( f(x) = \frac{\sqrt[3]{x}}{3} \). Here, \( f \) is the name of the function, and \( x \) is the input variable. Once you find its inverse, you label it \( f^{-1}(x) \), signifying the original function's reverse process.

Function notation not only helps keep track of what equations represent but also makes it clearer when you're dealing with different relationships, especially when solving and verifying inverses. It maintains clarity, especially in more complex calculus or algebra problems where multiple steps and functions are involved.

Using a proper notation system in math is not just about accuracy; it aids in communication, making your work readable and understandable to others.