Problem 51 - Calculus of a Single Variable

Question

Determine whether the function is one-to-one. If it is, find its inverse function. \(f(x)=\sqrt{x-2}\)

Step-by-Step Solution

Answer

The function \(f(x)=\sqrt{x-2}\) is one-to-one and its inverse function is \(f^{-1}(x)=x^{2}+2\) for \(x>=0\).

Step 1: Test for one-to-one property

To determine if \(f(x)=\sqrt{x-2}\) is one-to-one, we can visually examine the graph. This function is strictly increasing for all values of \(x\) in its domain. Thus, \(f(x)=\sqrt{x-2}\) is a one-to-one function since there is exactly one value of \(x\) for each unique value of \(f(x)\).

Step 2: Finding the Inverse

To find the inverse of the function, we start by replacing \(f(x)\) with \(y\), giving \(y=\sqrt{x-2}\). Next, we swap \(x\) and \(y\) to give \(x=\sqrt{y-2}\). Solving for \(y\) gives \(y=x^{2}+2\). Therefore, \(f^{-1}(x)=x^{2}+2\).

Step 3: Check the Validity of the Inverse

Due to the nature of the function involving a square root, we must check the validity of the inverse. Here, \(f^{-1}(x)=x^{2}+2\) where \(x>=0\). Therefore, the inverse function does exist for \(x>=0\) giving \(f^{-1}(x)=x^{2}+2\).

Key Concepts

Inverse Function

Understanding the concept of an inverse function is pivotal when dealing with mathematical functions. An inverse function, denoted as \(f^{-1}(x)\), is a function that reverses the mappings of the original function \(f\). This means that if you have a pairing in the original function where \(f(a) = b\), the inverse function will have the pairing \(f^{-1}(b) = a\). However, not all functions have an inverse.

For a function to have an inverse, it must be one-to-one, which implies that for every \(y\) value in the function's range, there is a unique \(x\) value associated with it. To find an inverse, we typically follow a set of steps: begin by replacing \(f(x)\) with \(y\), swap \(x\) and \(y\), and then solve for the new \(y\). It is crucial to validate the inverse to ensure it still constitutes a function with a domain corresponding to the original function's range. This step is particularly important for functions that have a restricted domain, such as square root functions.

Square Root Function

The square root function is a type of radical function defined by \(f(x)=\sqrt{x}\), which yields the principal or non-negative square root of \(x\). In the exercise \(f(x)=\sqrt{x-2}\), the function represents a shifted square root function, which means the graph and properties are modified according to the transformation.

The domain of \(f(x)=\sqrt{x-2}\) is \(x\geq2\), since the radicand (the expression under the square root) must be non-negative for the function to yield real numbers. This function increases as \(x\) increases, never decreases, which means it passes the horizontal line test and is one-to-one. When working with square root functions, always remember that their domains are restricted by the necessity of having a non-negative radicand, which is essential when finding inverses and ensuring the function's integrity.

Function Domain

The domain of a function consists of all the possible input values (\(x\)-values) that will yield a valid output from the function. It's one of the fundamental concepts that helps define a function's capabilities and limits. For example, in the square root function from our exercise, \(f(x)=\sqrt{x-2}\), the domain cannot include any \(x\)-values that make the expression under the square root negative, as that would result in imaginary numbers.

Thus, the domain for \(f(x)\) is \(x\geq2\), implying that you cannot substitute any value less than 2 for \(x\). When determining the domain for more complex functions, we consider all restrictions such as division by zero and even roots of negative numbers. It is crucial to analyze the domain before solving an equation or determining the inverse, as it impacts the validity and range of the resulting function.