- Set the function \(y = x^2 - 16\) with the domain \(x \geq 0\).
- Swap the variables \(x\) and \(y\) to obtain \(x = y^2 - 16\).
- Solve the equation for \(y\) while considering the domain restriction, resulting in \(y = \sqrt{x + 16}\).
- Identify the correct option as \(\boxed{f^{-1}(x) = \sqrt{x + 16}}\).
Explanation
- Defining the function and its domain
To find the inverse of the function \(f(x) = x^2 - 16\) with the domain \(x \geq 0\), we first set the function equal to a variable \(y\). So, we write \(y = x^2 - 16\). The condition \(x \geq 0\) is critical because the function \(f(x) = x^2 - 16\) is only a one-to-one mapping on this interval, which allows an inverse to exist. The range of this function is \([-16, \infty)\).
- Calculating the inverse function
To find the inverse, we swap the roles of \(x\) and \(y\). We start with the equation \(x = y^2 - 16\). Now, our goal is to solve for \(y\) in terms of \(x\). Adding 16 to both sides gives \(y^2 = x + 16\). Taking the square root of both sides, we get \(y = \pm\sqrt{x + 16}\). Since the original domain was \(x \geq 0\), the range of the inverse function must also be non-negative (\(y \geq 0\)). Therefore, we must choose the positive root, yielding \(y = \sqrt{x + 16}\).
- Conclusion and final answer
Comparing our result \(f^{-1}(x) = \sqrt{x + 16}\) with the provided options, we see that it corresponds to the first option. Thus, the inverse of \(f(x) = x^2 - 16\) for \(x \geq 0\) is \(f^{-1}(x) = \sqrt{x + 16}\). The final answer is \(\boxed{f^{-1}(x) = \sqrt{x + 16}}\).
Examples
Inverse functions have many practical applications, such as in finance and engineering. For example, if you have a formula that calculates the area of a circle based on its radius, \(A(r) = \pi r^2\), finding the inverse allows you to calculate the radius required to achieve a specific area, \(r(A) = \sqrt{\frac{A}{\pi}}\). Similarly, understanding the inverse of a function like \(f(x) = x^2 - 16\) helps in situations where you need to reverse a physical process modeled by a quadratic equation, such as determining the time it takes for an object to reach a specific altitude or finding an input value that resulted in a known output.