Problem 67 - 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. $$ f(x)=x^{2 / 3}+1 $$

Step-by-Step Solution

Answer

Restrict domain to \( x \geq 0 \), inverse is \( f^{-1}(x) = (x-1)^{3/2} \).

Step 1: Analyze the Function

The given function is \( f(x) = x^{2/3} + 1 \). This function is not one-to-one on the entire real line because it does not pass the horizontal line test due to its symmetry about the y-axis. We need to restrict the domain to make it one-to-one.

Step 2: Restrict the Domain

To make \( f(x) = x^{2/3} + 1 \) one-to-one, we can restrict the domain to \( x \geq 0 \) (only the right side of the y-axis). On this domain, the function is increasing and therefore satisfies the one-to-one property.

Step 3: Write the Inverse Function

First, solve for \( x \) in terms of \( y \):1. Start from the equation \( y = x^{2/3} + 1 \).2. Subtract 1 from both sides: \( y - 1 = x^{2/3} \).3. Solve for \( x \) by cubing both sides: \( x = (y - 1)^{3/2} \).Hence, the inverse function is \( f^{-1}(x) = (x - 1)^{3/2} \).

Step 4: State the Inverse Domain

Since the original function \( f(x) \) was restricted to \( x \geq 0 \), the domain of the inverse function \( f^{-1}(x) = (x - 1)^{3/2} \) will be \( x \geq 1 \) because the range of the original function becomes the domain of the inverse.

Key Concepts

Domain Restriction

Domain restriction is a crucial step in ensuring that a function becomes one-to-one, especially when dealing with functions like quadratics, which are typically not one-to-one due to symmetry. In this exercise, the function given is \( f(x) = x^{2/3} + 1 \). By its nature, this function is even and symmetric about the y-axis, meaning it does not naturally lend itself to one-to-one behavior.

To restrict the domain, we need to limit the portion of the function we consider. Here, we look at the domain where \( x \geq 0 \). This choice makes sense because, on this domain, the function is monotonically increasing (constantly increasing), eliminating the potential for any horizontal line to intersect the graph more than once. By considering only \( x \geq 0 \), we ensure that our function meets the criteria it needs to be one-to-one.

Horizontal Line Test

The horizontal line test is a simple, visual method to determine if a function is one-to-one. A function is one-to-one if no horizontal line cuts through its graph at more than one point.

Let's consider our function \( f(x) = x^{2/3} + 1 \). Without any domain restrictions, draw several horizontal lines at various values of \( y \). You'll quickly notice that each horizontal line interacts with the graph more than once when considering the complete domain. This observation reveals the function is not one-to-one.

• To pass the horizontal line test, most functions need a domain restriction. For \( f(x) \), restricting \( x \geq 0 \) ensures any horizontal line will only touch the graph once, confirming the function's one-to-one status.

One-to-One Function

A one-to-one function is such that each output is paired with exactly one input. This property is essential when finding the inverse of a function since every output of the inverse must map back to a unique input of the original function.

In our example, once we apply the domain restriction of \( x \geq 0 \) to \( f(x) = x^{2/3} + 1 \), the function becomes one-to-one. This restriction means as \( x \) increases, \( f(x) \) steadily increases without repeating any values. Thus, every \( y \)-value has a unique \( x \) from \( x \geq 0 \).

Knowing a function is one-to-one allows us to proceed confidently in finding the inverse. The inverse function is \( f^{-1}(x) = (x - 1)^{3/2} \), effectively reversing the roles of outputs and inputs, supported by our domain restriction which ensures the one-to-one nature.