Problem 68 - College Algebra with Modeling and Visualization

Question

Restrict the domain of \(f(x)\) so that \(f\) is one to-one. Then find \(f^{-1}(x)\). Answers may vary. $$ f(x)=2(x+3)^{2 / 3} $$

Step-by-Step Solution

Answer

Restrict the domain to \(x \geq -3\) and the inverse is \(f^{-1}(x) = \left( \frac{x}{2} \right)^{3/2} - 3\).

Step 1: Analyze the Function

The function given is \( f(x) = 2(x+3)^{2/3} \). Before restricting the domain, observe that the expression \((x+3)^{2/3}\) is not one-to-one as it involves an even root, which does not pass the Horizontal Line Test.

Step 2: Determine the Appropriate Domain Restriction

To make the function one-to-one, restrict the domain where the expression \((x+3)\) is non-negative. Choose \(x \geq -3\) for the domain of \(f\). This ensures \((x+3)^{2/3}\) is always non-negative and makes \(f\) one-to-one.

Step 3: Find the Inverse Function

To find the inverse, start by setting \(y = f(x) = 2(x+3)^{2/3}\). Solve for \(x\) in terms of \(y\):\[ y = 2(x+3)^{2/3} \]Divide both sides by 2:\[ \frac{y}{2} = (x+3)^{2/3} \]Next, raise both sides to the power of \(3/2\) to isolate \(x+3\):\[ \left( \frac{y}{2} \right)^{3/2} = x+3 \]Finally, solve for \(x\):\[ x = \left( \frac{y}{2} \right)^{3/2} - 3 \]Thus, the inverse function is \( f^{-1}(x) = \left( \frac{x}{2} \right)^{3/2} - 3 \).

Key Concepts

Domain Restriction

Domain restriction is an important concept when dealing with inverse functions, particularly when a function is not naturally one-to-one. To make a function one-to-one, we sometimes limit its domain to a specific range where it behaves in a one-to-one manner.
In the given exercise, the function is \( f(x) = 2(x+3)^{2/3} \). By default, this function is not one-to-one because it contains an even root, \((x+3)^{2/3}\), which affects its behavior across the entire set of real numbers. A function is one-to-one if every output corresponds to exactly one input.
To restrict the domain, we look for values of \(x\) where the function behaves monotonically, that is, either always increasing or always decreasing. Here, choosing \(x \geq -3\) ensures that \((x+3)\) remains non-negative, simplifying the expression and ensuring it passes the Horizontal Line Test. This domain restriction makes it possible to derive an inverse later on.

Horizontal Line Test

The Horizontal Line Test is a visual way to determine whether a function is one-to-one. When you graph a function, draw horizontal lines across the graph. If any horizontal line intersects the graph at more than one point, the function is not one-to-one in that domain.
For the function \( f(x) = 2(x+3)^{2/3} \), we need to verify whether it can be made one-to-one through a domain restriction. Initially, it doesn't pass the Horizontal Line Test because values of \(x\) that result in \((x+3)\) crossing negative values will produce the same outputs due to the even power root.
However, once we restrict the domain to \(x \geq -3\), the function becomes one-to-one, as the even root operation becomes well-defined and positive, ensuring a unique output for each input. So, passing the test is crucial to finding the inverse function successfully.

One-to-One Function

A one-to-one function is an essential property required for a function to have an inverse. In simpler terms, each input has a unique output and vice versa, which means no two different input values \(x_1\) and \(x_2\) will produce the same output \(f(x_1) eq f(x_2)\).
The original function \( f(x) = 2(x+3)^{2/3} \) is not naturally one-to-one over its entire domain because reflecting properties of the even root allow two different \(x\) values to provide the same result. However, by limiting the domain to \(x \geq -3\), you prevent overlapping outputs, transforming it into a one-to-one function.
Once the function has this property, calculating the inverse is achievable by switching \(x\) and \(y\) and solving for the former in terms of the latter. This results in the invert function \( f^{-1}(x) = \left( \frac{x}{2} \right)^{3/2} - 3 \), which can be verified for its application in real-world problems that require reversible processes.