- Determine the domain (\(x \geq 5\)) and range (\(y \geq 0\)) of the function \(f(x) = \sqrt{x - 5}\).
- Set \(y = \sqrt{x - 5}\) and swap \(x\) and \(y\) to obtain \(x = \sqrt{y - 5}\).
- Square both sides to get \(x^2 = y - 5\) and solve for \(y\), resulting in \(y = x^2 + 5\).
- State the domain of the inverse function as the original range, so the inverse is \(f^{-1}(x) = x^2 + 5\) for \(x \geq 0\), which corresponds to option A.
- Final Answer: \(\boxed{A}\)
Explanation
- Analysis of the Function and its Properties
The given function is \(f(x) = \sqrt{x - 5}\). To find its inverse, we first determine the domain and range of the function. For the square root to be defined, the expression inside must be non-negative: \(x - 5 \geq 0\), which means the domain is \(x \geq 5\). Since the output of the square root is always non-negative, the range of the function is \(y \geq 0\). The inverse function exists because the original function is one-to-one on its domain.
- Finding the Inverse Function
To find the inverse, we set \(y = f(x)\), so \(y = \sqrt{x - 5}\). We then interchange \(x\) and \(y\) to solve for the inverse, resulting in the equation \(x = \sqrt{y - 5}\). Note that since \(x\) was the input for the original function, now it must be in the domain of the inverse, which is the range of the original function, so \(x \geq 0\). To isolate \(y\), we square both sides: \(x^2 = y - 5\). Adding 5 to both sides gives \(y = x^2 + 5\). Thus, the inverse function is \(f^{-1}(x) = x^2 + 5\), defined for \(x \geq 0\). Comparing this result with the given options, we identify it as option A.
Examples
In many real-world scenarios, finding an inverse function is essential for reversing a process. For example, if you have a formula that calculates the amount of time \(t\) required to travel a distance \(d\) at a certain speed, the inverse function would allow you to calculate the distance \(d\) given the time \(t\). Understanding how to swap variables and solve for the original input helps in converting between different units, reversing economic models, or decoding encrypted data, making it a foundational skill in algebra and beyond.